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数学研发论坛»论坛 › 【数学研究】 › 小题大作 › 数字串的通项公式
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楼主: 王守恩

[原创] 数字串的通项公式

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王守恩
 楼主| 发表于 2026-3-9 17:17:02 | 显示全部楼层
接楼上。

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 2], {n, 2, 9}, {m, n + 1, 40 + 2 n}]
{{3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946}, ——C(n,2,2)
{3, 4, 6, 7, 11, 12, 17, 19, 24, 26, 33, 35, 43, 46, 54, 57, 67, 70, 81, 85, 96, 100, 113, 117, 131, 136, 150, 155, 171, 176, 193, 199, 216, 222, 241, 247, 267, 274, 294, 301, 323, 330, 353}——C(n,3,2)——A011975
{3, 3, 4, 6, 7, 8, 11, 12, 13, 18, 19, 20, 26, 27, 29, 35, 37, 39, 46, 48, 50, 59, 61, 63, 73, 75, 78, 88, 91, 94, 105, 108, 111, 124, 127, 130, 144, 147, 151, 165, 169, 173, 188, 192},——C(n,4,2)——A011976
{3, 3, 4, 4, 6, 7, 8, 8, 12, 12, 13, 14, 18, 19, 20, 21, 27, 28, 29, 30, 37, 38, 40, 41, 48, 50, 52, 53, 62, 63, 65, 67, 76, 78, 80, 82, 93, 95, 97, 99, 111, 113, 116, 118, 130}, ——C(n,5,2)——A011977
{3, 3, 3, 4, 4, 6, 7, 7, 8, 8, 12, 12, 13, 14, 14, 19, 20, 20, 21, 22, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 50, 51, 52, 54, 55, 63, 65, 66, 68, 69, 79, 80, 82, 84, 85, 96}, ——C(n,6,2)
{3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 9, 12, 12, 13, 14, 14, 15, 19, 20, 20, 21, 22, 23, 28, 29, 30, 30, 31, 32, 38, 39, 40, 41, 42, 43, 51, 52, 53, 54, 55, 56, 65, 66, 67, 69, 70}, ——C(n,7,2)
{3, 3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 19, 20, 20, 21, 22, 22, 23, 28, 29, 30, 30, 31, 32, 33, 39, 40, 41, 42, 42, 43, 44, 51, 52, 53, 54, 55, 56}, ——C(n,8,2)
{3, 3, 3, 3, 4, 4, 4, 4, 6, 7, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 15, 19, 20, 20, 21, 22, 22, 23, 23, 28, 29, 30, 30, 31, 32, 32, 33, 39, 40, 41, 42, 42, 43, 44, 45, 52}}——C(n,9,2)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)]], {n, 2, 9}, {m, n + 1, 40 + 2 n}]
{{3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946}, ——C(n,2,2)
{3, 4, 6, 7, 11, 12, 17, 19, 24, 26, 33, 35, 43, 46, 54, 57, 67, 70, 81, 85, 96, 100, 113, 117, 131, 136, 150, 155, 171, 176, 193, 199, 216, 222, 241, 247, 267, 274, 294, 301, 323, 330, 353}——C(n,3,2)——A011975
{3, 3, 4, 6, 7, 8, 11, 12, 13, 18, 19, 20, 26, 27, 29, 35, 37, 39, 46, 48, 50, 59, 61, 63, 73, 75, 78, 88, 91, 94, 105, 108, 111, 124, 127, 130, 144, 147, 151, 165, 169, 173, 188, 192}, ——C(n,4,2)——A011976
{3, 3, 4, 4, 6, 7, 8, 8, 12, 12, 13, 14, 18, 19, 20, 21, 27, 28, 29, 30, 37, 38, 40, 41, 48, 50, 52, 53, 62, 63, 65, 67, 76, 78, 80, 82, 93, 95, 97, 99, 111, 113, 116, 118, 130}, ——C(n,5,2)——A011977
{3, 3, 3, 4, 4, 6, 7, 7, 8, 8, 12, 12, 13, 14, 14, 19, 20, 20, 21, 22, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 50, 51, 52, 54, 55, 63, 65, 66, 68, 69, 79, 80, 82, 84, 85, 96}, ——C(n,6,2)
{3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 9, 12, 12, 13, 14, 14, 15, 19, 20, 20, 21, 22, 23, 28, 29, 30, 30, 31, 32, 38, 39, 40, 41, 42, 43, 51, 52, 53, 54, 55, 56, 65, 66, 67, 69, 70}, ——C(n,7,2)
{3, 3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 19, 20, 20, 21, 22, 22, 23, 28, 29, 30, 30, 31, 32, 33, 39, 40, 41, 42, 42, 43, 44, 51, 52, 53, 54, 55, 56}, ——C(n,8,2)
{3, 3, 3, 3, 4, 4, 4, 4, 6, 7, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 15, 19, 20, 20, 21, 22, 22, 23, 23, 28, 29, 30, 30, 31, 32, 32, 33, 39, 40, 41, 42, 42, 43, 44, 45, 52}}——C(n,9,2)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 3], {n, 3, 9}, {m, n + 1, 40 + n}]
{{4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341},
{4, 6, 11, 14, 25, 30, 47, 57, 78, 91, 124, 140, 183, 207, 257, 285, 352, 385, 466, 510, 600, 650, 763, 819, 950, 1020, 1163, 1240, 1411, 1496, 1689, 1791, 1998, 2109, 2350, 2470, 2737, 2877, 3161, 3311}, ——C(n,4,3)
{4, 5, 7, 11, 14, 18, 27, 32, 37, 54, 61, 68, 94, 103, 116, 147, 163, 180, 221, 240, 260, 319, 342, 366, 438, 465, 500, 581, 619, 658, 756, 800, 844, 968, 1016, 1066, 1210, 1265, 1329, 1485}, ——C(n,5,3)
{4, 4, 6, 7, 11, 14, 18, 19, 30, 32, 37, 42, 57, 64, 70, 77, 104, 112, 121, 130, 167, 178, 194, 205, 248, 267, 286, 301, 362, 378, 401, 425, 494, 520, 547, 574, 667, 697, 728, 759}, ——C(n,6,3)
{4, 4, 5, 7, 7, 12, 14, 15, 19, 20, 31, 33, 38, 42, 44, 63, 69, 72, 78, 85, 108, 116, 125, 133, 142, 180, 190, 200, 211, 222, 272, 285, 298, 317, 330, 387, 409, 425, 447, 464},——C(n,7,3)
{4, 4, 5, 6, 7, 7, 12, 14, 15, 18, 19, 23, 32, 33, 38, 42, 44, 49, 65, 70, 73, 79, 86, 92, 116, 124, 132, 135, 144, 152, 186, 195, 205, 216, 226, 237, 287, 299, 312, 324}, ——C(n,8,3)
{4, 4, 4, 5, 7, 7, 8, 12, 14, 15, 18, 19, 20, 23, 32, 34, 38, 39, 44, 46, 50, 66, 72, 74, 80, 86, 88, 95, 119, 126, 134, 137, 145, 153, 162, 195, 205, 215, 224, 229}}——C(n,9,3)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1) Ceiling[(m - 2)/(n - 2)]]], {n, 3, 9}, {m, n + 1, 40 + n}]
{{4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341},
{4, 6, 11, 14, 25, 30, 47, 57, 78, 91, 124, 140, 183, 207, 257, 285, 352, 385, 466, 510, 600, 650, 763, 819, 950, 1020, 1163, 1240, 1411, 1496, 1689, 1791, 1998, 2109, 2350, 2470, 2737, 2877, 3161, 3311}, ——C(n,4,3)
{4, 5, 7, 11, 14, 18, 27, 32, 37, 54, 61, 68, 94, 103, 116, 147, 163, 180, 221, 240, 260, 319, 342, 366, 438, 465, 500, 581, 619, 658, 756, 800, 844, 968, 1016, 1066, 1210, 1265, 1329, 1485}, ——C(n,5,3)
{4, 4, 6, 7, 11, 14, 18, 19, 30, 32, 37, 42, 57, 64, 70, 77, 104, 112, 121, 130, 167, 178, 194, 205, 248, 267, 286, 301, 362, 378, 401, 425, 494, 520, 547, 574, 667, 697, 728, 759}, ——C(n,6,3)
{4, 4, 5, 7, 7, 12, 14, 15, 19, 20, 31, 33, 38, 42, 44, 63, 69, 72, 78, 85, 108, 116, 125, 133, 142, 180, 190, 200, 211, 222, 272, 285, 298, 317, 330, 387, 409, 425, 447, 464}, ——C(n,7,3)
{4, 4, 5, 6, 7, 7, 12, 14, 15, 18, 19, 23, 32, 33, 38, 42, 44, 49, 65, 70, 73, 79, 86, 92, 116, 124, 132, 135, 144, 152, 186, 195, 205, 216, 226, 237, 287, 299, 312, 324}, ——C(n,8,3)
{4, 4, 4, 5, 7, 7, 8, 12, 14, 15, 18, 19, 20, 23, 32, 34, 38, 39, 44, 46, 50, 66, 72, 74, 80, 86, 88, 95, 119, 126, 134, 137, 145, 153, 162, 195, 205, 215, 224, 229}}——C(n,9,3)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 4], {n, 4, 9}, {m, n + 1, 47}]
{{5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270},
{5, 9, 18, 26, 50, 66, 113, 149, 219, 273, 397, 476, 659, 787, 1028, 1197, 1549, 1771, 2237, 2550, 3120, 3510, 4273, 4751, 5700, 6324, 7444, 8184, 9595, 10472, 12161, 13254, 15185, 16451, 18800, 20254, 22991, 24743, 27817, 29799},
{5, 7, 11, 19, 26, 36, 59, 75, 93, 144, 173, 204, 298, 344, 406, 539, 625, 720, 921, 1040, 1170, 1489, 1653, 1830, 2263, 2480, 2750, 3293, 3611, 3948, 4662, 5067, 5486, 6454, 6943, 7462, 8672, 9277, 9968, 11385, 12181}, ——C(n,6,4)
{5, 6, 9, 11, 19, 26, 36, 41, 69, 78, 96, 114, 163, 192, 220, 253, 357, 400, 450, 502, 668, 738, 832, 908, 1134, 1259, 1390, 1505, 1862, 1998, 2177, 2368, 2823, 3046, 3282, 3526, 4193, 4481, 4784, 5097}, ——C(n,7,4)
{5, 5, 7, 11, 12, 21, 27, 30, 41, 45, 74, 83, 100, 116, 127, 189, 216, 234, 264, 298, 392, 435, 485, 532, 586, 765, 832, 900, 976, 1055, 1326, 1425, 1528, 1665, 1774, 2129, 2301, 2444, 2627}, ——C(n,8,4)
{5, 5, 7, 9, 11, 12, 22, 27, 30, 38, 43, 54, 79, 85, 102, 117, 128, 147, 203, 226, 244, 273, 306, 338, 439, 483, 528, 555, 608, 659, 827, 889, 957, 1032, 1105, 1185, 1467, 1562}}——C(n,9,4)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)Ceiling[(m - 2)/(n - 2) Ceiling[(m - 3)/(n - 3)]]]], {n, 4, 9}, {m, n + 1, 47}]
{{5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270},
{5, 9, 18, 26, 50, 66, 113, 149, 219, 273, 397, 476, 659, 787, 1028, 1197, 1549, 1771, 2237, 2550, 3120, 3510, 4273, 4751, 5700, 6324, 7444, 8184, 9595, 10472, 12161, 13254, 15185, 16451, 18800, 20254, 22991, 24743, 27817, 29799},
{5, 7, 11, 19, 26, 36, 59, 75, 93, 144, 173, 204, 298, 344, 406, 539, 625, 720, 921, 1040, 1170, 1489, 1653, 1830, 2263, 2480, 2750, 3293, 3611, 3948, 4662, 5067, 5486, 6454, 6943, 7462, 8672, 9277, 9968, 11385, 12181}, ——C(n,6,4)
{5, 6, 9, 11, 19, 26, 36, 41, 69, 78, 96, 114, 163, 192, 220, 253, 357, 400, 450, 502, 668, 738, 832, 908, 1134, 1259, 1390, 1505, 1862, 1998, 2177, 2368, 2823, 3046, 3282, 3526, 4193, 4481, 4784, 5097}, ——C(n,7,4)
{5, 5, 7, 11, 12, 21, 27, 30, 41, 45, 74, 83, 100, 116, 127, 189, 216, 234, 264, 298, 392, 435, 485, 532, 586, 765, 832, 900, 976, 1055, 1326, 1425, 1528, 1665, 1774, 2129, 2301, 2444, 2627}, ——C(n,8,4)
{5, 5, 7, 9, 11, 12, 22, 27, 30, 38, 43, 54, 79, 85, 102, 117, 128, 147, 203, 226, 244, 273, 306, 338, 439, 483, 528, 555, 608, 659, 827, 889, 957, 1032, 1105, 1185, 1467, 1562}}——C(n,9,4)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 5], {n, 5, 9}, {m, n + 1, 43}]
{{6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757},
{6, 12, 27, 44, 92, 132, 245, 348, 548, 728, 1125, 1428, 2087, 2624, 3598, 4389, 5938, 7084, 9321, 11050, 14040, 16380, 20653, 23755, 29450, 33728, 40942, 46376, 55971, 62832, 74993, 83942, 98703, 109674, 128467, 141778, 164769},
{6, 9, 16, 30, 45, 67, 118, 161, 213, 350, 445, 554, 852, 1032, 1276, 1771, 2143, 2572, 3421, 4012, 4680, 6169, 7085, 8105, 10346, 11692, 13358, 16465, 18571, 20868, 25308, 28231, 31349, 37802, 41658, 45838}, ——C(n,7,5)
{6, 8, 13, 17, 31, 46, 68, 82, 147, 176, 228, 285, 428, 528, 633, 759, 1116, 1300, 1519, 1757, 2422, 2768, 3224, 3632, 4678, 5351, 6082, 6773, 8612, 9491, 10613, 11840, 14468, 15992, 17641}, ——C(n,8,5)
{6, 7, 10, 16, 19, 35, 48, 57, 82, 95, 165, 194, 245, 297, 339, 525, 624, 702, 822, 961, 1307, 1499, 1725, 1951, 2214, 2975, 3328, 3700, 4121, 4572, 5894, 6492, 7131, 7955}}——C(n,9,5)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)Ceiling[(m - 2)/(n - 2)Ceiling[(m - 3)/(n - 3) Ceiling[(m - 4)/(n - 4)]]]]], {n, 5, 9}, {m, n + 1, 43}]
{{6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757},
{6, 12, 27, 44, 92, 132, 245, 348, 548, 728, 1125, 1428, 2087, 2624, 3598, 4389, 5938, 7084, 9321, 11050, 14040, 16380, 20653, 23755, 29450, 33728, 40942, 46376, 55971, 62832, 74993, 83942, 98703, 109674, 128467, 141778, 164769},
{6, 9, 16, 30, 45, 67, 118, 161, 213, 350, 445, 554, 852, 1032, 1276, 1771, 2143, 2572, 3421, 4012, 4680, 6169, 7085, 8105, 10346, 11692, 13358, 16465, 18571, 20868, 25308, 28231, 31349, 37802, 41658, 45838}, ——C(n,7,5)
{6, 8, 13, 17, 31, 46, 68, 82, 147, 176, 228, 285, 428, 528, 633, 759, 1116, 1300, 1519, 1757, 2422, 2768, 3224, 3632, 4678, 5351, 6082, 6773, 8612, 9491, 10613, 11840, 14468, 15992, 17641}, ——C(n,8,5)
{6, 7, 10, 16, 19, 35, 48, 57, 82, 95, 165, 194, 245, 297, 339, 525, 624, 702, 822, 961, 1307, 1499, 1725, 1951, 2214, 2975, 3328, 3700, 4121, 4572, 5894, 6492, 7131, 7955}}——C(n,9,5)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 6], {n, 6, 9}, {m, n + 1, 40}]
{{7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681},
{7, 16, 39, 70, 158, 246, 490, 746, 1253, 1768, 2893, 3876, 5963, 7872, 11308, 14421, 20359, 25300, 34621, 42622, 56160, 67860, 88513, 105201, 134629, 159004, 198862, 231880, 287851, 332112, 407105, 467677, 564018}, ——C(n,7,6)
{7, 12, 22, 45, 74, 118, 222, 322, 453, 788, 1057, 1385, 2237, 2838, 3669, 5313, 6697, 8359, 11546, 14042, 16965, 23134, 27455, 32420, 42678, 49691, 58442, 74093, 85891, 99123, 123377, 141155},——C(n,8,6)
{7, 10, 18, 25, 49, 77, 121, 155, 294, 372, 507, 665, 1047, 1350, 1688, 2109, 3224, 3900, 4726, 5662, 8074, 9535, 11464, 13318, 17673, 20810, 24328, 27845, 36362, 41128, 47169}}——C(n,9,6)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)Ceiling[(m - 2)/(n - 2)Ceiling[(m - 3)/(n - 3)Ceiling[(m - 4)/(n - 4) Ceiling[(m - 5)/(n - 5)]]]]]], {n, 6, 9}, {m, n + 1, 40}]
{{7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681},
{7, 16, 39, 70, 158, 246, 490, 746, 1253, 1768, 2893, 3876, 5963, 7872, 11308, 14421, 20359, 25300, 34621, 42622, 56160, 67860, 88513, 105201, 134629, 159004, 198862, 231880, 287851, 332112, 407105, 467677, 564018}, ——C(n,7,6)
{7, 12, 22, 45, 74, 118, 222, 322, 453, 788, 1057, 1385, 2237, 2838, 3669, 5313, 6697, 8359, 11546, 14042, 16965, 23134, 27455, 32420, 42678, 49691, 58442, 74093, 85891, 99123, 123377, 141155}, ——C(n,8,6)
{7, 10, 18, 25, 49, 77, 121, 155, 294, 372, 507, 665, 1047, 1350, 1688, 2109, 3224, 3900, 4726, 5662, 8074, 9535, 11464, 13318, 17673, 20810, 24328, 27845, 36362, 41128, 47169}}——C(n,9,6)

公式(3)。Table[Fold[Ceiling[(#1 (m - #2))/(n - #2)] &, 1, Range[5, 0, -1]], {n, 6, 9}, {m, n + 1, 40}]
{{7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681},
{7, 16, 39, 70, 158, 246, 490, 746, 1253, 1768, 2893, 3876, 5963, 7872, 11308, 14421, 20359, 25300, 34621, 42622, 56160, 67860, 88513, 105201, 134629, 159004, 198862, 231880, 287851, 332112, 407105, 467677, 564018}, ——C(n,7,6)
{7, 12, 22, 45, 74, 118, 222, 322, 453, 788, 1057, 1385, 2237, 2838, 3669, 5313, 6697, 8359, 11546, 14042, 16965, 23134, 27455, 32420, 42678, 49691, 58442, 74093, 85891, 99123, 123377, 141155}, ——C(n,8,6)
{7, 10, 18, 25, 49, 77, 121, 155, 294, 372, 507, 665, 1047, 1350, 1688, 2109, 3224, 3900, 4726, 5662, 8074, 9535, 11464, 13318, 17673, 20810, 24328, 27845, 36362, 41128, 47169}}——C(n,9,6)
       
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王守恩
 楼主| 发表于 2026-3-10 07:35:27 | 显示全部楼层
接楼上。

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 2], {n, 2, 9}, {m, n + 1, 40 + 2 n}]
{{3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946}, ——C(n,2,2)
{3, 4, 6, 7, 11, 12, 17, 19, 24, 26, 33, 35, 43, 46, 54, 57, 67, 70, 81, 85, 96, 100, 113, 117, 131, 136, 150, 155, 171, 176, 193, 199, 216, 222, 241, 247, 267, 274, 294, 301, 323, 330, 353}——C(n,3,2)——A011975
{3, 3, 4, 6, 7, 8, 11, 12, 13, 18, 19, 20, 26, 27, 29, 35, 37, 39, 46, 48, 50, 59, 61, 63, 73, 75, 78, 88, 91, 94, 105, 108, 111, 124, 127, 130, 144, 147, 151, 165, 169, 173, 188, 192},——C(n,4,2)——A011976
{3, 3, 4, 4, 6, 7, 8, 8, 12, 12, 13, 14, 18, 19, 20, 21, 27, 28, 29, 30, 37, 38, 40, 41, 48, 50, 52, 53, 62, 63, 65, 67, 76, 78, 80, 82, 93, 95, 97, 99, 111, 113, 116, 118, 130}, ——C(n,5,2)——A011977
{3, 3, 3, 4, 4, 6, 7, 7, 8, 8, 12, 12, 13, 14, 14, 19, 20, 20, 21, 22, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 50, 51, 52, 54, 55, 63, 65, 66, 68, 69, 79, 80, 82, 84, 85, 96}, ——C(n,6,2)
{3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 9, 12, 12, 13, 14, 14, 15, 19, 20, 20, 21, 22, 23, 28, 29, 30, 30, 31, 32, 38, 39, 40, 41, 42, 43, 51, 52, 53, 54, 55, 56, 65, 66, 67, 69, 70}, ——C(n,7,2)
{3, 3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 19, 20, 20, 21, 22, 22, 23, 28, 29, 30, 30, 31, 32, 33, 39, 40, 41, 42, 42, 43, 44, 51, 52, 53, 54, 55, 56}, ——C(n,8,2)
{3, 3, 3, 3, 4, 4, 4, 4, 6, 7, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 15, 19, 20, 20, 21, 22, 22, 23, 23, 28, 29, 30, 30, 31, 32, 32, 33, 39, 40, 41, 42, 42, 43, 44, 45, 52}}——C(n,9,2)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)]], {n, 2, 9}, {m, n + 1, 40 + 2 n}]
{{3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, 276, 300, 325, 351, 378, 406, 435, 465, 496, 528, 561, 595, 630, 666, 703, 741, 780, 820, 861, 903, 946}, ——C(n,2,2)
{3, 4, 6, 7, 11, 12, 17, 19, 24, 26, 33, 35, 43, 46, 54, 57, 67, 70, 81, 85, 96, 100, 113, 117, 131, 136, 150, 155, 171, 176, 193, 199, 216, 222, 241, 247, 267, 274, 294, 301, 323, 330, 353}——C(n,3,2)——A011975
{3, 3, 4, 6, 7, 8, 11, 12, 13, 18, 19, 20, 26, 27, 29, 35, 37, 39, 46, 48, 50, 59, 61, 63, 73, 75, 78, 88, 91, 94, 105, 108, 111, 124, 127, 130, 144, 147, 151, 165, 169, 173, 188, 192}, ——C(n,4,2)——A011976
{3, 3, 4, 4, 6, 7, 8, 8, 12, 12, 13, 14, 18, 19, 20, 21, 27, 28, 29, 30, 37, 38, 40, 41, 48, 50, 52, 53, 62, 63, 65, 67, 76, 78, 80, 82, 93, 95, 97, 99, 111, 113, 116, 118, 130}, ——C(n,5,2)——A011977
{3, 3, 3, 4, 4, 6, 7, 7, 8, 8, 12, 12, 13, 14, 14, 19, 20, 20, 21, 22, 27, 28, 29, 30, 31, 38, 39, 40, 41, 42, 50, 51, 52, 54, 55, 63, 65, 66, 68, 69, 79, 80, 82, 84, 85, 96}, ——C(n,6,2)
{3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 9, 12, 12, 13, 14, 14, 15, 19, 20, 20, 21, 22, 23, 28, 29, 30, 30, 31, 32, 38, 39, 40, 41, 42, 43, 51, 52, 53, 54, 55, 56, 65, 66, 67, 69, 70}, ——C(n,7,2)
{3, 3, 3, 3, 4, 4, 4, 6, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 19, 20, 20, 21, 22, 22, 23, 28, 29, 30, 30, 31, 32, 33, 39, 40, 41, 42, 42, 43, 44, 51, 52, 53, 54, 55, 56}, ——C(n,8,2)
{3, 3, 3, 3, 4, 4, 4, 4, 6, 7, 7, 7, 8, 8, 8, 9, 12, 12, 13, 13, 14, 14, 15, 15, 19, 20, 20, 21, 22, 22, 23, 23, 28, 29, 30, 30, 31, 32, 32, 33, 39, 40, 41, 42, 42, 43, 44, 45, 52}}——C(n,9,2)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 3], {n, 3, 9}, {m, n + 1, 40 + n}]
{{4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341},
{4, 6, 11, 14, 25, 30, 47, 57, 78, 91, 124, 140, 183, 207, 257, 285, 352, 385, 466, 510, 600, 650, 763, 819, 950, 1020, 1163, 1240, 1411, 1496, 1689, 1791, 1998, 2109, 2350, 2470, 2737, 2877, 3161, 3311}, ——C(n,4,3)
{4, 5, 7, 11, 14, 18, 27, 32, 37, 54, 61, 68, 94, 103, 116, 147, 163, 180, 221, 240, 260, 319, 342, 366, 438, 465, 500, 581, 619, 658, 756, 800, 844, 968, 1016, 1066, 1210, 1265, 1329, 1485}, ——C(n,5,3)
{4, 4, 6, 7, 11, 14, 18, 19, 30, 32, 37, 42, 57, 64, 70, 77, 104, 112, 121, 130, 167, 178, 194, 205, 248, 267, 286, 301, 362, 378, 401, 425, 494, 520, 547, 574, 667, 697, 728, 759}, ——C(n,6,3)
{4, 4, 5, 7, 7, 12, 14, 15, 19, 20, 31, 33, 38, 42, 44, 63, 69, 72, 78, 85, 108, 116, 125, 133, 142, 180, 190, 200, 211, 222, 272, 285, 298, 317, 330, 387, 409, 425, 447, 464},——C(n,7,3)
{4, 4, 5, 6, 7, 7, 12, 14, 15, 18, 19, 23, 32, 33, 38, 42, 44, 49, 65, 70, 73, 79, 86, 92, 116, 124, 132, 135, 144, 152, 186, 195, 205, 216, 226, 237, 287, 299, 312, 324}, ——C(n,8,3)
{4, 4, 4, 5, 7, 7, 8, 12, 14, 15, 18, 19, 20, 23, 32, 34, 38, 39, 44, 46, 50, 66, 72, 74, 80, 86, 88, 95, 119, 126, 134, 137, 145, 153, 162, 195, 205, 215, 224, 229}}——C(n,9,3)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1) Ceiling[(m - 2)/(n - 2)]]], {n, 3, 9}, {m, n + 1, 40 + n}]
{{4, 10, 20, 35, 56, 84, 120, 165, 220, 286, 364, 455, 560, 680, 816, 969, 1140, 1330, 1540, 1771, 2024, 2300, 2600, 2925, 3276, 3654, 4060, 4495, 4960, 5456, 5984, 6545, 7140, 7770, 8436, 9139, 9880, 10660, 11480, 12341},
{4, 6, 11, 14, 25, 30, 47, 57, 78, 91, 124, 140, 183, 207, 257, 285, 352, 385, 466, 510, 600, 650, 763, 819, 950, 1020, 1163, 1240, 1411, 1496, 1689, 1791, 1998, 2109, 2350, 2470, 2737, 2877, 3161, 3311}, ——C(n,4,3)
{4, 5, 7, 11, 14, 18, 27, 32, 37, 54, 61, 68, 94, 103, 116, 147, 163, 180, 221, 240, 260, 319, 342, 366, 438, 465, 500, 581, 619, 658, 756, 800, 844, 968, 1016, 1066, 1210, 1265, 1329, 1485}, ——C(n,5,3)
{4, 4, 6, 7, 11, 14, 18, 19, 30, 32, 37, 42, 57, 64, 70, 77, 104, 112, 121, 130, 167, 178, 194, 205, 248, 267, 286, 301, 362, 378, 401, 425, 494, 520, 547, 574, 667, 697, 728, 759}, ——C(n,6,3)
{4, 4, 5, 7, 7, 12, 14, 15, 19, 20, 31, 33, 38, 42, 44, 63, 69, 72, 78, 85, 108, 116, 125, 133, 142, 180, 190, 200, 211, 222, 272, 285, 298, 317, 330, 387, 409, 425, 447, 464}, ——C(n,7,3)
{4, 4, 5, 6, 7, 7, 12, 14, 15, 18, 19, 23, 32, 33, 38, 42, 44, 49, 65, 70, 73, 79, 86, 92, 116, 124, 132, 135, 144, 152, 186, 195, 205, 216, 226, 237, 287, 299, 312, 324}, ——C(n,8,3)
{4, 4, 4, 5, 7, 7, 8, 12, 14, 15, 18, 19, 20, 23, 32, 34, 38, 39, 44, 46, 50, 66, 72, 74, 80, 86, 88, 95, 119, 126, 134, 137, 145, 153, 162, 195, 205, 215, 224, 229}}——C(n,9,3)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 4], {n, 4, 9}, {m, n + 1, 47}]
{{5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270},
{5, 9, 18, 26, 50, 66, 113, 149, 219, 273, 397, 476, 659, 787, 1028, 1197, 1549, 1771, 2237, 2550, 3120, 3510, 4273, 4751, 5700, 6324, 7444, 8184, 9595, 10472, 12161, 13254, 15185, 16451, 18800, 20254, 22991, 24743, 27817, 29799},
{5, 7, 11, 19, 26, 36, 59, 75, 93, 144, 173, 204, 298, 344, 406, 539, 625, 720, 921, 1040, 1170, 1489, 1653, 1830, 2263, 2480, 2750, 3293, 3611, 3948, 4662, 5067, 5486, 6454, 6943, 7462, 8672, 9277, 9968, 11385, 12181}, ——C(n,6,4)
{5, 6, 9, 11, 19, 26, 36, 41, 69, 78, 96, 114, 163, 192, 220, 253, 357, 400, 450, 502, 668, 738, 832, 908, 1134, 1259, 1390, 1505, 1862, 1998, 2177, 2368, 2823, 3046, 3282, 3526, 4193, 4481, 4784, 5097}, ——C(n,7,4)
{5, 5, 7, 11, 12, 21, 27, 30, 41, 45, 74, 83, 100, 116, 127, 189, 216, 234, 264, 298, 392, 435, 485, 532, 586, 765, 832, 900, 976, 1055, 1326, 1425, 1528, 1665, 1774, 2129, 2301, 2444, 2627}, ——C(n,8,4)
{5, 5, 7, 9, 11, 12, 22, 27, 30, 38, 43, 54, 79, 85, 102, 117, 128, 147, 203, 226, 244, 273, 306, 338, 439, 483, 528, 555, 608, 659, 827, 889, 957, 1032, 1105, 1185, 1467, 1562}}——C(n,9,4)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)Ceiling[(m - 2)/(n - 2) Ceiling[(m - 3)/(n - 3)]]]], {n, 4, 9}, {m, n + 1, 47}]
{{5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365, 1820, 2380, 3060, 3876, 4845, 5985, 7315, 8855, 10626, 12650, 14950, 17550, 20475, 23751, 27405, 31465, 35960, 40920, 46376, 52360, 58905, 66045, 73815, 82251, 91390, 101270},
{5, 9, 18, 26, 50, 66, 113, 149, 219, 273, 397, 476, 659, 787, 1028, 1197, 1549, 1771, 2237, 2550, 3120, 3510, 4273, 4751, 5700, 6324, 7444, 8184, 9595, 10472, 12161, 13254, 15185, 16451, 18800, 20254, 22991, 24743, 27817, 29799},
{5, 7, 11, 19, 26, 36, 59, 75, 93, 144, 173, 204, 298, 344, 406, 539, 625, 720, 921, 1040, 1170, 1489, 1653, 1830, 2263, 2480, 2750, 3293, 3611, 3948, 4662, 5067, 5486, 6454, 6943, 7462, 8672, 9277, 9968, 11385, 12181}, ——C(n,6,4)
{5, 6, 9, 11, 19, 26, 36, 41, 69, 78, 96, 114, 163, 192, 220, 253, 357, 400, 450, 502, 668, 738, 832, 908, 1134, 1259, 1390, 1505, 1862, 1998, 2177, 2368, 2823, 3046, 3282, 3526, 4193, 4481, 4784, 5097}, ——C(n,7,4)
{5, 5, 7, 11, 12, 21, 27, 30, 41, 45, 74, 83, 100, 116, 127, 189, 216, 234, 264, 298, 392, 435, 485, 532, 586, 765, 832, 900, 976, 1055, 1326, 1425, 1528, 1665, 1774, 2129, 2301, 2444, 2627}, ——C(n,8,4)
{5, 5, 7, 9, 11, 12, 22, 27, 30, 38, 43, 54, 79, 85, 102, 117, 128, 147, 203, 226, 244, 273, 306, 338, 439, 483, 528, 555, 608, 659, 827, 889, 957, 1032, 1105, 1185, 1467, 1562}}——C(n,9,4)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 5], {n, 5, 9}, {m, n + 1, 43}]
{{6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757},
{6, 12, 27, 44, 92, 132, 245, 348, 548, 728, 1125, 1428, 2087, 2624, 3598, 4389, 5938, 7084, 9321, 11050, 14040, 16380, 20653, 23755, 29450, 33728, 40942, 46376, 55971, 62832, 74993, 83942, 98703, 109674, 128467, 141778, 164769},
{6, 9, 16, 30, 45, 67, 118, 161, 213, 350, 445, 554, 852, 1032, 1276, 1771, 2143, 2572, 3421, 4012, 4680, 6169, 7085, 8105, 10346, 11692, 13358, 16465, 18571, 20868, 25308, 28231, 31349, 37802, 41658, 45838}, ——C(n,7,5)
{6, 8, 13, 17, 31, 46, 68, 82, 147, 176, 228, 285, 428, 528, 633, 759, 1116, 1300, 1519, 1757, 2422, 2768, 3224, 3632, 4678, 5351, 6082, 6773, 8612, 9491, 10613, 11840, 14468, 15992, 17641}, ——C(n,8,5)
{6, 7, 10, 16, 19, 35, 48, 57, 82, 95, 165, 194, 245, 297, 339, 525, 624, 702, 822, 961, 1307, 1499, 1725, 1951, 2214, 2975, 3328, 3700, 4121, 4572, 5894, 6492, 7131, 7955}}——C(n,9,5)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)Ceiling[(m - 2)/(n - 2)Ceiling[(m - 3)/(n - 3) Ceiling[(m - 4)/(n - 4)]]]]], {n, 5, 9}, {m, n + 1, 43}]
{{6, 21, 56, 126, 252, 462, 792, 1287, 2002, 3003, 4368, 6188, 8568, 11628, 15504, 20349, 26334, 33649, 42504, 53130, 65780, 80730, 98280, 118755, 142506, 169911, 201376, 237336, 278256, 324632, 376992, 435897, 501942, 575757},
{6, 12, 27, 44, 92, 132, 245, 348, 548, 728, 1125, 1428, 2087, 2624, 3598, 4389, 5938, 7084, 9321, 11050, 14040, 16380, 20653, 23755, 29450, 33728, 40942, 46376, 55971, 62832, 74993, 83942, 98703, 109674, 128467, 141778, 164769},
{6, 9, 16, 30, 45, 67, 118, 161, 213, 350, 445, 554, 852, 1032, 1276, 1771, 2143, 2572, 3421, 4012, 4680, 6169, 7085, 8105, 10346, 11692, 13358, 16465, 18571, 20868, 25308, 28231, 31349, 37802, 41658, 45838}, ——C(n,7,5)
{6, 8, 13, 17, 31, 46, 68, 82, 147, 176, 228, 285, 428, 528, 633, 759, 1116, 1300, 1519, 1757, 2422, 2768, 3224, 3632, 4678, 5351, 6082, 6773, 8612, 9491, 10613, 11840, 14468, 15992, 17641}, ——C(n,8,5)
{6, 7, 10, 16, 19, 35, 48, 57, 82, 95, 165, 194, 245, 297, 339, 525, 624, 702, 822, 961, 1307, 1499, 1725, 1951, 2214, 2975, 3328, 3700, 4121, 4572, 5894, 6492, 7131, 7955}}——C(n,9,5)

公式(1)。S[m_, k_, t_] := Module[{W = 1, m1 = m, k1 = k, t1 = t}, m1 += 1 - t1; k1 += 1 - t1; While[t1 > 0, W = Ceiling[W*m1/k1]; t1--; m1++; k1++]; W]; Table[S[m, n, 6], {n, 6, 9}, {m, n + 1, 40}]
{{7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681},
{7, 16, 39, 70, 158, 246, 490, 746, 1253, 1768, 2893, 3876, 5963, 7872, 11308, 14421, 20359, 25300, 34621, 42622, 56160, 67860, 88513, 105201, 134629, 159004, 198862, 231880, 287851, 332112, 407105, 467677, 564018}, ——C(n,7,6)
{7, 12, 22, 45, 74, 118, 222, 322, 453, 788, 1057, 1385, 2237, 2838, 3669, 5313, 6697, 8359, 11546, 14042, 16965, 23134, 27455, 32420, 42678, 49691, 58442, 74093, 85891, 99123, 123377, 141155},——C(n,8,6)
{7, 10, 18, 25, 49, 77, 121, 155, 294, 372, 507, 665, 1047, 1350, 1688, 2109, 3224, 3900, 4726, 5662, 8074, 9535, 11464, 13318, 17673, 20810, 24328, 27845, 36362, 41128, 47169}}——C(n,9,6)

公式(2)。Table[Ceiling[m/n Ceiling[(m - 1)/(n - 1)Ceiling[(m - 2)/(n - 2)Ceiling[(m - 3)/(n - 3)Ceiling[(m - 4)/(n - 4) Ceiling[(m - 5)/(n - 5)]]]]]], {n, 6, 9}, {m, n + 1, 40}]
{{7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681},
{7, 16, 39, 70, 158, 246, 490, 746, 1253, 1768, 2893, 3876, 5963, 7872, 11308, 14421, 20359, 25300, 34621, 42622, 56160, 67860, 88513, 105201, 134629, 159004, 198862, 231880, 287851, 332112, 407105, 467677, 564018}, ——C(n,7,6)
{7, 12, 22, 45, 74, 118, 222, 322, 453, 788, 1057, 1385, 2237, 2838, 3669, 5313, 6697, 8359, 11546, 14042, 16965, 23134, 27455, 32420, 42678, 49691, 58442, 74093, 85891, 99123, 123377, 141155}, ——C(n,8,6)
{7, 10, 18, 25, 49, 77, 121, 155, 294, 372, 507, 665, 1047, 1350, 1688, 2109, 3224, 3900, 4726, 5662, 8074, 9535, 11464, 13318, 17673, 20810, 24328, 27845, 36362, 41128, 47169}}——C(n,9,6)

公式(3)。Table[Fold[Ceiling[(#1 (m - #2))/(n - #2)] &, 1, Range[5, 0, -1]], {n, 6, 9}, {m, n + 1, 40}]
{{7, 28, 84, 210, 462, 924, 1716, 3003, 5005, 8008, 12376, 18564, 27132, 38760, 54264, 74613, 100947, 134596, 177100, 230230, 296010, 376740, 475020, 593775, 736281, 906192, 1107568, 1344904, 1623160, 1947792, 2324784, 2760681},
{7, 16, 39, 70, 158, 246, 490, 746, 1253, 1768, 2893, 3876, 5963, 7872, 11308, 14421, 20359, 25300, 34621, 42622, 56160, 67860, 88513, 105201, 134629, 159004, 198862, 231880, 287851, 332112, 407105, 467677, 564018}, ——C(n,7,6)
{7, 12, 22, 45, 74, 118, 222, 322, 453, 788, 1057, 1385, 2237, 2838, 3669, 5313, 6697, 8359, 11546, 14042, 16965, 23134, 27455, 32420, 42678, 49691, 58442, 74093, 85891, 99123, 123377, 141155}, ——C(n,8,6)
{7, 10, 18, 25, 49, 77, 121, 155, 294, 372, 507, 665, 1047, 1350, 1688, 2109, 3224, 3900, 4726, 5662, 8074, 9535, 11464, 13318, 17673, 20810, 24328, 27845, 36362, 41128, 47169}}——C(n,9,6)

公式(4)。Table[Product[(m - a)/(n - a), {a, 0, n - 1}], {n, 9}, {m, n + 1, 40}]——C(n,a,a)——可以简化——公式(4)。

上面的所有数据≤正确数据。正确数据见——La Jolla Covering Repository Tables
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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王守恩
 楼主| 发表于 2026-3-11 06:49:09 | 显示全部楼层
A004080——Least k such that H(k) >= n, where H(k) is the harmonic number Sum_{i=1..k} 1/i.

1, 4, 11, 31, 83, 227, 616, 1674, 4550, 12367, 33617, 91380, 248397, 675214, 1835421, 4989191, 13562027, 36865412, 100210581, 272400600, 740461601, 2012783315, 5471312310, 14872568831, 40427833596, 109894245429,

Table[If[n == 1, 1, Round[Exp[n - EulerGamma]]], {n, 10000}]——这样就可以了！！！
毋因群疑而阻独见　　毋任己意而废人言
毋私小惠而伤大体　　毋借公论以快私情
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王守恩
 楼主| 发表于 2026-3-11 10:19:00 | 显示全部楼层
A081721——Number of bracelets of n beads in up to n colors.

{1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356, 24330595997127372497, 1092955780817066765469}

Table[CycleIndex[DihedralGroup[n], s]/.Table[s[i]->n, {i, 1, n}], {n, 1, 20}]——这个显示不了。

Table[If[n == 2, 3, CycleIndexPolynomial[DihedralGroup[n], Array[s, n]] /. s[_] -> n], {n, 20}]——这个就可以。

Table[DivisorSum[n, EulerPhi[#] n^(n/#) &]/(2 n) + (n^Ceiling[n/2] + n^Ceiling[(n + 1)/2])/4, {n, 20}]——这样也行。
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 楼主| 发表于 2026-3-18 07:41:43 | 显示全部楼层
A287106——Positions of 1 in A287104.
{1, 4, 6, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 36, 38, 41, 43, 46, 48, 50, 53, 55, 57, 59, 62, 64, 66, 69, 71, 73, 75, 78, 80, 83, 85, 87, 90, 92, 94, 96, 99, 101, 103, 106, 108, 111, 113, 115, 118, 120, 122, 124, 127, 129, 131, 134,
136, 138, 140, 143, 145, 148, 150, 152, 155, 157, 159, 161, 164, 166, 168, 171, 173, 176, 178, 180, 183, 185, 187, 189, 192, 194, 197, 199, 201, 204, 206, 208, 210, 213, 215, 217, 220, 222, 224, 226, 229, 231, 234, 236, 238, 241, 243,
245, 247, 250, 252, 254, 257, 259, 262, 264, 266, 269, 271, 273, 275, 278, 280, 283, 285, 287, 290, 292, 294, 296, 299, 301, 303, 306, 308, 311, 313, 315, 318, 320, 322, 324, 327, 329, 331, 334, 336, 338, 340, 343, 345, 348, 350}
Flatten[Position[NestWhile[Flatten[#/.{0->{1,0},1->{1,2},2->0}]&,{0},Count[#,1]<100&,1,∞],1]][[;;100]]——通项公式可以这样。
A287106——与下面的数字串没有关系。
{5, 8, 10, 13, 15, 17, 20, 22, 25, 27, 29, 32, 34, 36, 39, 41, 43, 45, 48, 50, 52, 55, 57, 59, 61, 64, 66, 68, 71, 73, 75, 77, 80, 82, 84, 86, 89, 91, 93, 95, 98, 100, 102, 104, 106, 109, 111, 113, 115, 118, 120, 122, 124, 127, 129, 131, 133, 135, 138,
140, 142, 144, 147, 149, 151, 153, 155, 158, 160, 162, 164, 167, 169, 171, 173, 175, 178, 180, 182, 184, 187, 189, 191, 193, 195, 198, 200, 202, 204, 206, 209, 211, 213, 215, 217, 220, 222, 224, 226, 228, 231, 233, 235, 237, 239, 242, 244, 246,
248, 250, 253, 255, 257, 259, 261, 264, 266, 268, 270, 272, 275, 277, 279, 281, 283, 286, 288, 290, 292, 294, 297, 299, 301, 303, 305, 308, 310, 312, 314, 316, 318, 321, 323, 325, 327, 329, 332, 334, 336, 338, 340, 343, 345, 347, 349, 351}
Table[Ceiling[Log[n!, n^2!]], {n, 2, 157}]
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 楼主| 发表于 2026-3-25 10:30:03 | 显示全部楼层
A071095——Number of ways to tile hexagon of edges n, n+1, n+1, n, n+1, n+1 with diamonds of side 1.

1, 6, 175, 24696, 16818516, 55197331332, 872299918503728, 66345156372852988800, 24277282058281388285162560, 42730166102274086598901662210000, 361690697335823816369045433734882109375,

  1. Table[Product[i! (2 n + i)!/((n + i)!)^2, {i, 0, n - 2}], {n, 14}]——这样就行！！！！
复制代码

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 楼主| 发表于 2026-3-25 17:28:38 | 显示全部楼层
在7(列)×2(行)的方格内填1,2,3。每个方格恰好填1个数，要求左方格≤右方格,上方格≤下方格,有540种填法。
从简单开始。
1(列)×2(行)有001+001+01+01+1+1=006种填法。
2(列)×2(行)有006+005+03+03+2+1=020种填法。
3(列)×2(行)有020+014+06+06+3+1=050种填法。
4(列)×2(行)有050+030+10+10+4+1=105种填法。
5(列)×2(行)有105+055+15+15+5+1=196种填法。
6(列)×2(行)有196+091+21+21+6+1=336种填法。
7(列)×2(行)有336+140+28+28+7+1=540种填法。

\(\cdots\cdots\)\(\cdots\cdots\)

A002415——gives the number of 2*2 arrays that can be populated with 0..n such that rows and columns are nondecreasing.
6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156, 58870, 67425, 76880, 87296, 98736,- 埃里克·W·韦斯坦，2026年3月14日

在a(列)×b(行)的方格内填1,2,3,...,c。每个方格恰好填1个数，要求左方格≤右方格,上方格≤下方格,有S(a,b,c)种填法。
S(a,2,3){6, 20, 50, 105, 196, 336, 540, 825, 1210, 1716, 2366, 3185, 4200, 5440, 6936, 8721, 10830, 13300, 16170, 19481, 23276, 27600, 32500, 38025, 44226, 51156,58870, 67425}——A002415,
S(a,3,3){10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575}——A006542,
S(a,4,3){15, 105, 490, 1764, 5292, 13860, 32670, 70785, 143143, 273273, 496860, 866320, 1456560, 2372112, 3755844, 5799465, 8756055, 12954865, 18818646, 26883780, 37823500, 52474500}——A006857,
S(a,5,3){21, 196, 1176, 5292, 19404, 60984, 169884, 429429, 1002001, 2186184, 4504864, 8836464, 16604784, 30046752, 52581816, 89311761, 147685461, 238369516, 376372920, 582481900, 885069900}——A108679,
S(a,6,3){28, 336, 2520, 13860, 60984, 226512, 736164, 2147145, 5725720, 14158144, 32821152, 71954064, 150233760, 300467520, 578399976, 1075994073, 1941008916, 3405278800, 5824819000, 9735768900}——A134288,
S(a,2,4){10, 50, 175, 490, 1176, 2520, 4950, 9075, 15730, 26026, 41405, 63700, 95200, 138720, 197676, 276165, 379050, 512050, 681835, 896126, 1163800, 1495000, 1901250, 2395575, 2992626,}——A006542,
S(a,3,4){20, 175, 980, 4116, 14112, 41580, 108900, 259545, 572572, 1184183, 2318680, 4331600, 7768320, 13441968, 22535064, 36729945, 58373700, 90684055, 138003404, 206108980, 302588000}——A047819,
S(a,4,4){35, 490, 4116, 24696, 116424, 457380, 1557270, 4723719, 13026013, 33157124, 78835120, 176729280, 376375104, 766192176, 1498581756, 2828205765, 5168991135, 9177226366, 15870391460}——A107915,
S(a,5,4){56, 1176, 14112, 116424, 731808, 3737448, 16195608, 61408347, 208416208, 644195552, 1837984512, 4892876352, 12259074816, 29115302688, 65937597264, 143107211709, 298915373064, 603074875480}——A140901,
S(a,6,4){84, 2520, 41580, 457380, 3737448, 24293412, 131589315, 614083470, 2530768240, 9386849472, 31803696288, 99604982880, 291153026880, 800670823920, 2085276513474, 5172303508911, 12276881393700}——A140903,

\(\cdots\cdots\)\(\cdots\cdots\)

可以有统一的通项公式——\(\D S(a,b,c)=\prod_{i=1}^a\prod_{j=1}^b\frac{i + j + c - 2}{i + j - 1}\) ——OEIS可没有这么干脆的！！！
毋因群疑而阻独见　　毋任己意而废人言
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王守恩
 楼主| 发表于 2026-3-26 06:45:03 | 显示全部楼层
接楼上——在a(列)×b(行)的方格内填1,2,3,...,c。每个方格恰好填1个数，要求左方格≤右方格,上方格≤下方格,有S(a,b,c)种填法。

\(\D S(a,b,c)=\prod_{i=1}^a\prod_{j=1}^b\frac{i + j + c - 2}{i + j - 1}\) —— 双重乘积——OEIS可没有这么干脆的！！！

  1. 双重乘积代码是这样——Table[Product[(i + j + c - 2)/(i + j - 1), {i, a}, {j, b}], {c, 3, 4}, {b, 2, 6}, {a, 28}]
复制代码
  1. 单重乘积代码是这样——Table[Product[((s + c - 2)/(s - 1))^Min[a, b, s - 1, a + b + 1 - s], {s, 2, a + b}], {c, 3, 4}, {b, 2, 6}, {a, 28}]
复制代码

\(\D S(a,b,c)=\prod_{s=2}^{a+b}\bigg(\frac{s + c - 2}{s - 1}\bigg)^k\)  其中:  k = Min[a, b, s - 1, a + b + 1 - s]——单重乘积—— 这个不好看！！！
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 楼主| 发表于 2026-3-28 06:31:28 | 显示全部楼层
{1, 8, 70, 694, 6932, 69316, 693148, 6931473, 69314719, 693147181, 6931471806, 69314718057}——OEIS没有这串数。——Ceiling[Log[2]/Log[1 + 1/10^Range[0, 19]]]
a(0)=1, (1+1/10^0)^1≥2。
a(1)=8, (1+1/10^1)^8≥2。
a(2)=70, (1+1/10^2)^70≥2。
a(3)=694, (1+1/10^3)^694≥2。
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 楼主| 发表于 2026-4-2 07:39:40 | 显示全部楼层
解不定方程x^2+2y^2=3z^2。

Solve[{x^2 + 2 y^2 == 3 z^2, x > 0, y > 0, 20 > z > 0}, {z, y, x}, Integers]——给出基本解。
{{z -> 1, y -> 1, x -> 1}, {z -> 2, y -> 2, x -> 2}, {z -> 3, y -> 1, x -> 5}, {z -> 3, y -> 3, x -> 3}, {z -> 4, y -> 4, x -> 4}, {z -> 5, y -> 5, x -> 5}, {z -> 6, y -> 2, x -> 10},
{z -> 6, y -> 6, x -> 6}, {z -> 7, y -> 7, x -> 7}, {z -> 8, y -> 8, x -> 8}, {z -> 9, y -> 3, x -> 15}, {z -> 9, y -> 9, x -> 9}, {z -> 9, y -> 11, x -> 1}, {z -> 10, y -> 10, x -> 10},
{z -> 11, y -> 1, x -> 19}, {z -> 11, y -> 11, x -> 11}, {z -> 11, y -> 13, x -> 5}, {z -> 12, y -> 4, x -> 20}, {z -> 12, y -> 12, x -> 12}, {z -> 13, y -> 13, x -> 13}, {z -> 14, y -> 14, x -> 14},
{z -> 15, y -> 5, x -> 25}, {z -> 15, y -> 15, x -> 15}, {z -> 16, y -> 16, x -> 16}, {z -> 17, y -> 11, x -> 25}, {z -> 17, y -> 13, x -> 23}, {z -> 17, y -> 17, x -> 17}, {z -> 18, y -> 6, x -> 30},
{z -> 18, y -> 18, x -> 18}, {z -> 18, y -> 22, x -> 2}, {z -> 19, y -> 11, x -> 29}, {z -> 19, y -> 19, x -> 19}, {z -> 19, y -> 23, x -> 5}}

Solve[{x^2 + 2 y^2 == 3 z^2, x > 0, y > 0, 30 > z > 0, x ≠ y}, {z, y, x}, Integers]——增加 x ≠ y。
{{z -> 3, y -> 1, x -> 5}, {z -> 6, y -> 2, x -> 10}, {z -> 9, y -> 3, x -> 15}, {z -> 9, y -> 11, x -> 1}, {z -> 11, y -> 1, x -> 19}, {z -> 11, y -> 13, x -> 5}, {z -> 12, y -> 4, x -> 20},
{z -> 15, y -> 5, x -> 25}, {z -> 17, y -> 11, x -> 25}, {z -> 17, y -> 13, x -> 23}, {z -> 18, y -> 6, x -> 30}, {z -> 18, y -> 22, x -> 2}, {z -> 19, y -> 11, x -> 29}, {z -> 19, y -> 23, x -> 5},
{z -> 21, y -> 7, x -> 35}, {z -> 22, y -> 2, x -> 38}, {z -> 22, y -> 26, x -> 10}, {z -> 24, y -> 8, x -> 40}, {z -> 27, y -> 9, x -> 45}, {z -> 27, y -> 13, x -> 43}, {z -> 27, y -> 33, x -> 3}}

Solve[{x^2 + 2 y^2 == 3 z^2, x > 0, y > 0, 50 > z > 0, x ≠ y, Mod[z, 3] ≠ 0}, {z, y, x}, Integers]——再增加 Mod[z, 3] ≠ 0。
{{z -> 11, y -> 1, x -> 19}, {z -> 11, y -> 13, x -> 5}, {z -> 17, y -> 11, x -> 25}, {z -> 17, y -> 13, x -> 23}, {z -> 19, y -> 11, x -> 29}, {z -> 19, y -> 23, x -> 5},
{z -> 22, y -> 2, x -> 38}, {z -> 22, y -> 26, x -> 10}, {z -> 34, y -> 22, x -> 50}, {z -> 34, y -> 26, x -> 46}, {z -> 38, y -> 22, x -> 58}, {z -> 38, y -> 46, x -> 10},
{z -> 41, y -> 1, x -> 71}, {z -> 41, y -> 47, x -> 25}, {z -> 43, y -> 23, x -> 67}, {z -> 43, y -> 37, x -> 53}, {z -> 44, y -> 4, x -> 76}, {z -> 44, y -> 52, x -> 20}}

Select[Solve[{x^2 + 2 y^2 == 3 z^2, x > 0, y > 0, 80 > z > 1, Mod[z, 3] ≠ 0}, {z, y, x}, Integers], GCD[x /. #, y /. #, z /. #] == 1 &]——再增加 GCD[x, y, z] = 1。
{{z -> 11, y -> 1, x -> 19}, {z -> 11, y -> 13, x -> 5}, {z -> 17, y -> 11, x -> 25}, {z -> 17, y -> 13, x -> 23}, {z -> 19, y -> 11, x -> 29}, {z -> 19, y -> 23, x -> 5},
{z -> 41, y -> 1, x -> 71}, {z -> 41, y -> 47, x -> 25}, {z -> 43, y -> 23, x -> 67}, {z -> 43, y -> 37, x -> 53}, {z -> 59, y -> 11,  x -> 101}, {z -> 59, y -> 71, x -> 19},
{z -> 67, y -> 11, x -> 115}, {z -> 67, y -> 73, x -> 53}, {z -> 73, y -> 59, x -> 95}, {z -> 73, y -> 83, x -> 47}}

Union[z /. Select[Solve[{x^2 + 2 y^2 == 3 z^2, x > 0, y > 0, 1 < z < 400, Mod[z, 3] ≠ 0}, {z, y, x}, Integers], GCD[x /. #, y /. #, z /. #] == 1 &]]——单独把 z 提出来。
{11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 121, 131, 137, 139, 163, 179, 187, 193, 209, 211, 227, 233, 241, 251, 257, 281, 283, 289, 307, 313, 323, 331, 337, 347, 353, 361, 379}

Select[Range[1000], Mod[#, 3] != 0 && AllTrue[FactorInteger[#][[All, 1]], MemberQ[{1, 3}, Mod[#, 8]] &] &]——还是这串数——提速。——OEIS没有这串数。
{1, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 121, 131, 137, 139, 163, 179, 187, 193, 209, 211, 227, 233, 241, 251, 257, 281, 283, 289, 307, 313, 323, 331, 337, 347, 353, 361, 379, 401, 409, 419, 433, 443, 449, 451, 457, 467,
473, 491, 499, 521, 523, 547, 563, 569, 571, 577, 587, 593, 601, 617, 619, 641, 643, 649, 659, 673, 683, 691, 697, 731, 737, 739, 761, 769, 779, 787, 803, 809, 811, 817, 827, 857, 859, 881, 883, 907, 913, 929, 937, 947, 953, 971, 977, 979}

A033200——Primes congruent to {1, 3} (mod 8); or, odd primes of form x^2 + 2*y^2.——A033200有类似的数字串。
{3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 131, 137, 139, 163, 179, 193, 211, 227, 233, 241, 251, 257, 281, 283, 307, 313, 331, 337, 347, 353, 379, 401, 409, 419, 433, 443, 449, 457, 467, 491, 499, 521, 523, 547, 563, 569, 571, 577,
587, 593, 601, 617, 619, 641, 643, 659, 673, 683, 691, 739, 761, 769, 787, 809, 811, 827, 857, 859, 881, 883, 907, 929, 937, 947, 953, 971, 977, 1009, 1019, 1033, 1049, 1051, 1091, 1097, 1123, 1129, 1153, 1163, 1171, 1187, 1193, 1201, 1217}
A033200——Select[Prime[Range[200]], MemberQ[{1, 3}, Mod[#, 8]] &]——A033200的通项公式没有我们的好。

补充。
1,题目做不完, 思路是可以渗透的。
2,抓纲举目。本题, z 是关键。没有限制, z可以是任意数。1^2+2×1^2=3×1^2。
  z不是2的倍数,z不是3的倍数。z只能=11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107,113, 121,并且每个z恰好有2组解。
3,一个诱人的推论——A个数/B个数 = E/Pi 。
A={1, 3, 11, 17, 19, 41, 43, 59, 67, 73, 83, 89, 97, 107, 113, 121, 131, 137, 139, 163, 179, 187,
B={5, 7, 13, 23, 25, 29, 31, 35, 37, 47, 49, 53, 61, 65, 71, 79, 91, 101, 103, 109, 115, 125, 127,
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