数字串的通项公式

northwolves

northwolves 发表于 2025-3-10 12:48
{49,98,147,169,196,245,294,338,343,392,441,490,507,539,588,637,676,686,735,784,833,845,882,931,9 ...

数据有漏解：

1. s=Module[{n=2500},Union@Flatten@Table[r=(b^2+b+1)^2;r*Range[n/r],{b,2,n^(1/4)}]];{Length@s,s}

复制代码

{68,{49,98,147,169,196,245,294,338,343,392,441,490,507,539,588,637,676,686,735,784,833,845,882,931,961,980,1014,1029,1078,1127,1176,1183,1225,1274,1323,1352,1372,1421,1470,1519,1521,1568,1617,1666,1690,1715,1764,1813,1849,1859,1862,1911,1922,1960,2009,2028,2058,2107,2156,2197,2205,2254,2303,2352,2366,2401,2450,2499}}

1. t=Module[{n=2500},Union@Flatten@Table[r=(a^2+a b+b^2)^2;r*Range[n/r],{a,2,n^(1/4)},{b,a-1}]];{Length@t,t}

复制代码

{75,{49,98,147,169,196,245,294,338,343,361,392,441,490,507,539,588,637,676,686,722,735,784,833,845,882,931,961,980,1014,1029,1078,1083,1127,1176,1183,1225,1274,1323,1352,1369,1372,1421,1444,1470,1519,1521,1568,1617,1666,1690,1715,1764,1805,1813,1849,1859,1862,1911,1922,1960,2009,2028,2058,2107,2156,2166,2197,2205,2254,2303,2352,2366,2401,2450,2499}}

遗漏项：{361,722,1083,1369,1444,1805,2166}

361        {{36,100,225}}
722        {{72,200,450}}
1083        {{108,300,675}}
1369        {{144,441,784}}
1444        {{144,400,900}}
1805        {{180,500,1125}}
2166        {{216,600,1350}}

王守恩

northwolves 发表于 2025-3-11 20:14
数据有漏解：

{1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, 67, 92, 122, 163, 214, 279, 359, 462, 586, 744, 932, 1166, 1446, 1790, 2199, 2693, 3278, 3980, 4805, 5789, 6935, 8291, 9868, 11717, 13859, 16353, 19227, 22558, 26380}

1. Table[Length@Solve[{a + b + c + d + e + f + g + h + j + s + 5 k == n, 0 < a <= b <= c <= d <= e <= f <= g <= h <= j <= s, 0 <= k}, {k, a, b, c, d, e, f, g, h, j, s}, Integers], {n,10, 65}]

复制代码

可有好一点的通项公式？谢谢！！！

northwolves

王守恩 发表于 2025-3-17 08:11
{1, 1, 2, 3, 5, 8, 12, 17, 25, 35, 50, 67, 92, 122, 163, 214, 279, 359, 462, 586, 744, 932, 1166,  ...

1. Table[Sum[Length@IntegerPartitions[n-5k,{10}],{k,0,n/5}],{n,10,65}]

复制代码

{1,1,2,3,5,8,12,17,25,35,50,67,92,122,163,214,279,359,462,586,744,932,1166,1446,1790,2199,2693,3278,3980,4805,5789,6935,8291,9868,11717,13859,16353,19227,22558,26380,30787,35819,41594,48166,55672,64188,73866,84808,97203,111166,126928,144626,164533,186826,211820,239732}

王守恩

备忘——这样简单些。

Table[Sum[Sum[k, {a, k, (n - k)/2}], {k, n/3}], {n, 50}]
{0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658, 707, 763, 812, 868, 932, 988, 1052, 1124, 1188——A028289

Table[Sum[Sum[Sum[k, {a, b, (n - b - k)/2}], {b, k, (n - k)/3}], {k, n/4}], {n, 50}]
{0, 0, 0, 1, 1, 2, 3, 6, 7, 11, 14, 21, 25, 34, 41, 55, 64, 81, 95, 119, 136, 165, 189, 227, 256, 301, 339, 396, 441, 507, 564, 645, 711, 804,  885,  996, 1089, 1215, 1326, 1474, 1600, 1766, 1914, 2106, 2272, 2486, 2678——A308733

Table[Sum[Sum[Sum[Sum[k, {a, b, (n - b - c - k)/2}], {b, c, (n - c - k)/3}], {c, k, (n - k)/4}], {k, n/5}], {n, 50}]
{0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916,1050,1198, 1365, 1545, 1747, 1968, 2212, 2480, 2771, 3089, 3437, 3814, 4227, 4669, 5151——A308823

Table[Sum[Sum[Sum[Sum[Sum[k, {a, b, (n - b - c - d - k)/2}], {b, c, (n - c - d - k)/3}], {c, d, (n - d - k)/4}], {d, k, (n - k)/5}], {k, n/6}], {n, 50}]
{0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 12, 15, 22, 29, 40, 51, 70, 86, 112, 139, 176, 214, 269, 321, 394, 470, 567, 668, 801, 933,1103, 1281, 1498, 1725, 2007, 2293, 2643, 3010, 3443, 3897, 4439, 4995, 5652, 6341, 7135, 7967——A308868

Table[Sum[Sum[Sum[Sum[Sum[Sum[k, {a, b, (n - b - c - d - e - k)/2}], {b, c, (n - c - d - e - k)/3}], {c, d, (n - d - e - k)/4}], {d, e, (n - e - k)/5}], {e, k, (n - k)/6}], {k, n/7}], {n, 50}]
{0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 30, 41, 54, 72, 93, 121, 153, 194, 242, 302, 372, 457, 557,  675, 812, 975, 1162, 1381, 1632, 1924, 2254, 2636, 3068, 3562, 4119, 4752, 5463, 6265, 7162, 8170, 9293, 10549——A308927

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k, {a, b, (n - b - c - d - e - f - k)/2}], {b, c, (n - c - d - e - f - k)/3}], {c, d, (n - d - e - f - k)/4}], {d, e, (n - e - f - k)/5}], {e, f, (n - f - k)/6}], {f, k, (n - k)/7}], {k, n/8}], {n, 50}]
{0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 23, 30, 42, 55, 75, 96, 127,  161, 209, 260, 330, 407, 509, 621,  765,  925, 1128, 1350, 1627, 1934, 2310, 2725, 3227, 3782, 4447, 5178, 6044, 7000, 8122, 9355,  10791,  12370——A308990

Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k, {a, b, (n - b - c - d - e - f - g - k)/2}],{b, c, (n - c - d - e - f - g - k)/3}],{c, d, (n - d - e - f - g - k)/4}],{d, e, (n - e - f - g - k)/5}],{e, f, (n - f - g - k)/6}],{f, g, (n - g - k)/7}],{g, k, (n - k)/8}],{k, n/9}]
{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 42, 56, 76, 99, 130, 168, 216, 274, 349, 435, 544, 674, 831, 1017, 1244, 1507, 1823, 2194, 2629, 3136, 3734, 4420, 5223, 6148, 7215, 8438, 9851, 11453, 13292, 15382——A326465

王守恩

northwolves 发表于 2025-3-17 09:28
{1,1,2,3,5,8,12,17,25,35,50,67,92,122,163,214,279,359,462,586,744,932,1166,1446,1790,2199,2693,3 ...

两个的答案是一样的？谢谢！

Table[Total[IntegerPartitions[n, {2 k + 1}][[;; , 2 k + 1]]], {k, 5}, {n, 40}]
{{0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658},
{0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916, 1050, 1198, 1365, 1545, 1747, 1968, 2212, 2480},
{0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 30, 41, 54, 72, 93, 121, 153, 194, 242, 302, 372, 457, 557, 675, 812, 975, 1162, 1381, 1632, 1924, 2254, 2636, 3068, 3562, 4119},
{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 42, 56, 76, 99, 130, 168, 216, 274, 349, 435, 544, 674, 831, 1017, 1244, 1507, 1823, 2194, 2629, 3136, 3734, 4420},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 57, 77, 101, 134, 174, 226, 289, 370, 467, 590, 737, 920, 1137, 1404, 1720, 2104, 2557, 3101, 3740}}

Table[Total[IntegerPartitions[n, {2 k + 1}][[;; , -1]]], {k, 5}, {n, 40}]
{{0, 0, 1, 1, 2, 4, 5, 7, 11, 13, 17, 23, 27, 33, 42, 48, 57, 69, 78, 90, 106, 118, 134, 154, 170, 190, 215, 235, 260, 290, 315, 345, 381, 411, 447, 489, 525, 567, 616, 658},
{0, 0, 0, 0, 1, 1, 2, 3, 5, 8, 11, 15, 21, 28, 38, 48, 62, 78, 98, 122, 149, 181, 219, 262, 314, 370, 436, 510, 595, 691, 797, 916, 1050, 1198, 1365, 1545, 1747, 1968, 2212, 2480},
{0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 16, 22, 30, 41, 54, 72, 93, 121, 153, 194, 242, 302, 372, 457, 557, 675, 812, 975, 1162, 1381, 1632, 1924, 2254, 2636, 3068, 3562, 4119},
{0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 31, 42, 56, 76, 99, 130, 168, 216, 274, 349, 435, 544, 674, 831, 1017, 1244, 1507, 1823, 2194, 2629, 3136, 3734, 4420},
{0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 57, 77, 101, 134, 174, 226, 289, 370, 467, 590, 737, 920, 1137, 1404, 1720, 2104, 2557, 3101, 3740}}

王守恩

northwolves 发表于 2025-3-17 09:28
{1,1,2,3,5,8,12,17,25,35,50,67,92,122,163,214,279,359,462,586,744,932,1166,1446,1790,2199,2693,3 ...

把n分成不同正整数的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
a(1)=0,
a(2)=0,
a(3)=1,
a(4)=2,
a(5)=3,
a(6)=4,
a(7)=6,
a(8)=8,
a(9)=10,
a(10)=12,
a(11)=14,
a(12)=16,
a(13)=19,
a(14)=22,
a(15)=25,
a(16)=28,
a(17)=31,
a(18)=34,
a(19)=37,
a(20)=40,
a(21)=44,
a(22)=48,
a(23)=52,
a(24)=56,
a(25)=60,

0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 31, 34, 37, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 146, 152, 158, 164, 170, 176, 182, 188, 194, 200, 206, 212, 218, 224

还是没有通项公式——OEIS也没有。

northwolves

王守恩 发表于 2025-3-18 16:52
把n分成不同正整数的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
a(1)=0,
a(2)=0,

1. a[n_]:=(t=Floor[Sqrt[n+1/4]-1/2];n t-1/3 t (1+t) (2+t));Array[a,200]

复制代码

{0,0,1,2,3,4,6,8,10,12,14,16,19,22,25,28,31,34,37,40,44,48,52,56,60,64,68,72,76,80,85,90,95,100,105,110,115,120,125,130,135,140,146,152,158,164,170,176,182,188,194,200,206,212,218,224,231,238,245,252,259,266,273,280,287,294,301,308,315,322,329,336,344,352,360,368,376,384,392,400,408,416,424,432,440,448,456,464,472,480,489,498,507,516,525,534,543,552,561,570,579,588,597,606,615,624,633,642,651,660,670,680,690,700,710,720,730,740,750,760,770,780,790,800,810,820,830,840,850,860,870,880,891,902,913,924,935,946,957,968,979,990,1001,1012,1023,1034,1045,1056,1067,1078,1089,1100,1111,1122,1133,1144,1156,1168,1180,1192,1204,1216,1228,1240,1252,1264,1276,1288,1300,1312,1324,1336,1348,1360,1372,1384,1396,1408,1420,1432,1444,1456,1469,1482,1495,1508,1521,1534,1547,1560,1573,1586,1599,1612,1625,1638,1651,1664,1677,1690}

northwolves

王守恩 发表于 2025-3-18 16:52
把n分成不同正整数的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
a(1)=0,
a(2)=0,

1. a[n_]:=(t=Floor[Sqrt[n+1/4]-1/2];n t-1/3 t (1+t) (2+t));Array[a,200]

复制代码

{0,0,1,2,3,4,6,8,10,12,14,16,19,22,25,28,31,34,37,40,44,48,52,56,60,64,68,72,76,80,85,90,95,100,105,110,115,120,125,130,135,140,146,152,158,164,170,176,182,188,194,200,206,212,218,224,231,238,245,252,259,266,273,280,287,294,301,308,315,322,329,336,344,352,360,368,376,384,392,400,408,416,424,432,440,448,456,464,472,480,489,498,507,516,525,534,543,552,561,570,579,588,597,606,615,624,633,642,651,660,670,680,690,700,710,720,730,740,750,760,770,780,790,800,810,820,830,840,850,860,870,880,891,902,913,924,935,946,957,968,979,990,1001,1012,1023,1034,1045,1056,1067,1078,1089,1100,1111,1122,1133,1144,1156,1168,1180,1192,1204,1216,1228,1240,1252,1264,1276,1288,1300,1312,1324,1336,1348,1360,1372,1384,1396,1408,1420,1432,1444,1456,1469,1482,1495,1508,1521,1534,1547,1560,1573,1586,1599,1612,1625,1638,1651,1664,1677,1690}

王守恩

northwolves 发表于 2025-3-18 22:14
{0,0,1,2,3,4,6,8,10,12,14,16,19,22,25,28,31,34,37,40,44,48,52,56,60,64,68,72,76,80,85,90,95,100, ...

把n分成不同正整数的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
{0, 0, 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 19, 22, 25, 28, 31, 34, 37, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 135, 140, 146, 152},
Table[Sum[Round[Sqrt[n] - 1], {n, k}], {k, 90}]

把n分成不同自然数(可以有0)的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
{1, 2, 4, 6, 8, 10, 13, 16, 19, 22, 25, 28, 32, 36, 40, 44, 48, 52, 56, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 116, 122, 128, 134, 140, 146, 152, 158, 164, 170, 176, 182},
Table[Sum[Round[Sqrt[n]], {n, k}], {k, 90}]

把n分成正整数的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
{0, 0, 1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90, 100, 110, 121, 132, 144, 156, 169, 182, 196, 210, 225, 240, 256, 272, 289, 306, 324, 342, 361, 380, 400},
Table[Floor[n^2/4], {n, 90}]

把n分成自然数(可以有0)的和，使得这些数两两之差(大数-小数)的和最大=a(n)。
n=n+0+0+0+...,  a(n)=(n-0)+(n-0)+(n-0)+...

王守恩

这样也是挺好的！

\(\D\sum_{x = 1}^n\ \frac{2(n-x)+3}{3^x}=\sum_{x = 1}^n\ \frac{2x+1}{3^{n-x+1}}\)

{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65}