三角形数的三角形图

王守恩 · 发表于 2025-9-3 08:39:13

northwolves 发表于 2025-9-1 14:41

顶点是三角形数的三角形图——图见8#——记"0"关于"n"的对称数为"a(n)"。譬如
a(1)=11,
a(2)=13,
a(3)=15,
a(4)=17,
a(5)=19,
a(6)=21,
a(7)=23,
a(8)=25,
a(9)=27,
a(10)=56,
a(11)=58,
a(12)=60,
a(13)=62,
a(14)=64,
a(15)=66,
a(16)=68,
......
得到一串数——11, 13, 15, 17, 19, 21, 23, 25, 27, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 137, 139, 141,...。a(n) = 2 n + 9 Round[Sqrt[2 (n + 1)]/3]^2。

OEIS没有这串数——A062708也没有这个图——你掂量掂量——我是丢了！

northwolves · 发表于 2025-9-3 20:21:12

n=25;pts=Table[k=Round@Sqrt@t;(-1)^k ({-1,1} (Abs[k^2-t]-k)+k^2-t-Mod[k,2])/2,{t,0,n^2-1}];
Graphics[{{Thickness[0.002],Line@pts},Table[{EdgeForm[Red],Yellow,Disk[pts[[i]],0.4],Black,Text[Style[i-1,9],pts[[i]]]},{i,n^2}]},PlotRange->{{-n,n},{-n,n}}/2,ImageSize->600]

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王守恩 · 发表于 2025-9-4 10:41:42

northwolves 发表于 2025-9-3 20:21

接楼上。记 "0" 关于 "n" 的对称数为 "a(n)"。譬如
a(1)=10,
a(2)=12,
a(3)=14,
a(4)=16,
a(5)=18,
a(6)=20,
a(7)=22,
a(8)=24,
a(9)=50,
a(10)=52,
a(11)=54,
a(12)=56,
a(13)=58,
a(14)=60,
a(15)=62,
a(16)=64,
......
得到一串数——10, 12, 14, 16, 18, 20, 22, 24, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 122, 124, 126, 128, 130, 132,...。a(n) = 2 n + 8 Round[Sqrt[(2 n + 1)/8]]^2 。

这里有一个有趣的规律。如果把 "0", "n", "a(n)" 这条直线拉出来。譬如

b(1)={0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277}
b(2)={0, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306}
b(3)={0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335}
b(4)={0, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364}
b(5)={0, 5, 18, 39, 68, 105, 150, 203, 264, 333, 410, 495, 588, 689, 798, 915, 1040, 1173, 1314, 1463, 1620, 1785, 1958, 2139, 2328, 2525, 2730, 2943, 3164, 3393}
b(6)={0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422}
b(7)={0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451}
b(8)={0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480}
b(9)={0, 9, 50, 123, 228, 365, 534, 735, 968, 1233, 1530, 1859, 2220, 2613, 3038, 3495, 3984, 4505, 5058, 5643, 6260, 6909, 7590, 8303, 9048, 9825, 10634, 11475, 12348, 13253}
......
这些数字串却可以有一个共同的通项公式。b(n)=3*b(n-1)-3b(n-2)+b(n-3)。——太诱人！

我们再看另外一个有趣的规律。——在每个b(n)前面添一个数。
b(1) = {7, 0, 1, 10}
b(2) = {6, 0, 2, 12}
b(3) = {5, 0, 3, 14}——用后面3个数能把b(n)这条直线拉出来。
b(4) = {4, 0, 4, 16}——用前面3个数也是能把b(n)这条直线拉出来的。
b(5) = {3, 0, 5, 18}
b(6) = {2, 0, 6, 20}
b(7) = {1, 0, 7, 22}
b(8) = {0, 0, 8, 24}
b(9) = {23, 0, 9, 50}

......
第1个数是这样一串数。——7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 47, 46, 45, 44, 43, 42, 41, 40,...。 b(n) = - n + 8 Round[Sqrt[(2 n + 1)/8]]^2 。

从0开始，每个数都有——恰好出现一次。——比第4个数还是有规律些。

{7, 6, 5, 4, 3, 2, 1, 0,
23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8,
47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24,
79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48,
119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80,
167, 166, 165, 164, 163, 162, 161, 160, 159, 158, 157, 156, 155, 154, 153, 152, 151, 150, 149, 148, 147, 146, 145, 144, 143, 142, 141, 140, 139, 138, 137, 136, 135, 134, 133, 132, 131, 130, 129, 128, 127, 126, 125, 124, 123, 122, 121, 120}
......

王守恩 · 发表于 2025-9-4 13:43:02

northwolves 发表于 2025-9-3 20:21

接楼上。记 "0" 关于 "n" 的对称数为 "a(n)"。譬如
a(1)=10,
a(2)=12,
a(3)=14,
a(4)=16,
a(5)=18,
a(6)=20,
a(7)=22,
a(8)=24,
a(9)=50,
a(10)=52,
a(11)=54,
a(12)=56,
a(13)=58,
a(14)=60,
a(15)=62,
a(16)=64,
......
得到一串数——10, 12, 14, 16, 18, 20, 22, 24, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 122, 124, 126, 128, 130, 132,...。a(n) = 2 n + 8 Round[Sqrt[(2 n + 1)/8]]^2 。

这里有一个有趣的规律。如果把 "0", "n", "a(n)" 这条直线拉出来。譬如

b(1)={0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277}
b(2)={0, 2, 12, 30, 56, 90, 132, 182, 240, 306, 380, 462, 552, 650, 756, 870, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3080, 3306}
b(3)={0, 3, 14, 33, 60, 95, 138, 189, 248, 315, 390, 473, 564, 663, 770, 885, 1008, 1139, 1278, 1425, 1580, 1743, 1914, 2093, 2280, 2475, 2678, 2889, 3108, 3335}
b(4)={0, 4, 16, 36, 64, 100, 144, 196, 256, 324, 400, 484, 576, 676, 784, 900, 1024, 1156, 1296, 1444, 1600, 1764, 1936, 2116, 2304, 2500, 2704, 2916, 3136, 3364}
b(5)={0, 5, 18, 39, 68, 105, 150, 203, 264, 333, 410, 495, 588, 689, 798, 915, 1040, 1173, 1314, 1463, 1620, 1785, 1958, 2139, 2328, 2525, 2730, 2943, 3164, 3393}
b(6)={0, 6, 20, 42, 72, 110, 156, 210, 272, 342, 420, 506, 600, 702, 812, 930, 1056, 1190, 1332, 1482, 1640, 1806, 1980, 2162, 2352, 2550, 2756, 2970, 3192, 3422}
b(7)={0, 7, 22, 45, 76, 115, 162, 217, 280, 351, 430, 517, 612, 715, 826, 945, 1072, 1207, 1350, 1501, 1660, 1827, 2002, 2185, 2376, 2575, 2782, 2997, 3220, 3451}
b(8)={0, 8, 24, 48, 80, 120, 168, 224, 288, 360, 440, 528, 624, 728, 840, 960, 1088, 1224, 1368, 1520, 1680, 1848, 2024, 2208, 2400, 2600, 2808, 3024, 3248, 3480}
b(9)={0, 9, 50, 123, 228, 365, 534, 735, 968, 1233, 1530, 1859, 2220, 2613, 3038, 3495, 3984, 4505, 5058, 5643, 6260, 6909, 7590, 8303, 9048, 9825, 10634, 11475, 12348, 13253}
......
这些数字串却可以有一个共同的通项公式。b(n)=3*b(n-1)-3b(n-2)+b(n-3)。——太诱人！

我们再看另外一个有趣的规律。——在每个b(n)前面添一个数。
b(1) = {7, 0, 1, 10}
b(2) = {6, 0, 2, 12}
b(3) = {5, 0, 3, 14}——用后面3个数能把b(n)这条直线拉出来。
b(4) = {4, 0, 4, 16}——用前面3个数也是能把b(n)这条直线拉出来的。
b(5) = {3, 0, 5, 18}
b(6) = {2, 0, 6, 20}
b(7) = {1, 0, 7, 22}
b(8) = {0, 0, 8, 24}
b(9) = {23, 0, 9, 50}

......
第1个数是这样一串数。——7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 47, 46, 45, 44, 43, 42, 41, 40,...。 b(n) = - n + 8 Round[Sqrt[(2 n + 1)/8]]^2 。

从0开始，每个数都有——恰好出现一次。——OEIS有这样的数字串吗？——没有！！！

{7, 6, 5, 4, 3, 2, 1, 0,
23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8,
47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24,
79, 78, 77, 76, 75, 74, 73, 72, 71, 70, 69, 68, 67, 66, 65, 64, 63, 62, 61, 60, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48,
119, 118, 117, 116, 115, 114, 113, 112, 111, 110, 109, 108, 107, 106, 105, 104, 103, 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90, 89, 88, 87, 86, 85, 84, 83, 82, 81, 80,
167, 166, 165, 164, 163, 162, 161, 160, 159, 158, 157, 156, 155, 154, 153, 152, 151, 150, 149, 148, 147, 146, 145, 144, 143, 142, 141, 140, 139, 138, 137, 136, 135, 134, 133, 132, 131, 130, 129, 128, 127, 126, 125, 124, 123, 122, 121, 120}
......

王守恩 · 发表于 2025-9-4 21:29:36

从0开始，每个数都有——恰好出现一次。——OEIS有这样的数字串吗？——有！！！——OEIS有数字串(0)——后面的就没有了！！！

数字串(0) = {0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52},
数字串(1) = {1, 0, 5, 4, 3, 2, 11, 10, 9, 8, 7, 6, 19, 18, 17, 16, 15, 14, 13, 12, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 55, 54, 53, 52, 51, 50},
数字串(2) = {2, 1, 0, 8, 7, 6, 5, 4, 3, 17, 16, 15, 14, 13, 12, 11, 10, 9, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 62, 61, 60},
数字串(3) = {3, 2, 1, 0, 11, 10, 9, 8, 7, 6, 5, 4, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 59, 58, 57, 56, 55, 54, 53, 52},
数字串(4) = {4, 3, 2, 1, 0, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32},
数字串(5) = {5, 4, 3, 2, 1, 0, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48},
数字串(6) = {6, 5, 4, 3, 2, 1, 0, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 69, 68, 67, 66, 65, 64},
数字串(7) = {7, 6, 5, 4, 3, 2, 1, 0, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33, 32, 31, 30, 29, 28, 27, 26, 25, 24},
数字串(8) = {8, 7, 6, 5, 4, 3, 2, 1, 0, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42, 41, 40, 39, 38, 37, 36, 35, 34, 33},
数字串(9) = {9, 8, 7, 6, 5, 4, 3, 2, 1, 0, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 59, 58, 57, 56, 55, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 44, 43, 42},
......

A061579——0, 2, 1, 5, 4, 3, 9, 8, 7, 6, 14, 13, 12, 11, 10, 20, 19, 18, 17, 16, 15, 27, 26, 25, 24, 23, 22, 21, 35, 34, 33, 32, 31, 30, 29, 28, 44, 43, 42, 41, 40, 39, 38, 37, 36, 54, 53, 52, 51, 50, 49, 48, 47, 46, 45, 65, 64, 63,

去看看——A061579——就知道我们的通项公式有多好了——a(n)=Round[Sqrt[2 n]]^2 - n,。

我们的数字串太多了。你应该去申报几个！

王守恩 · 发表于 2025-9-5 07:46:53

northwolves 发表于 2025-9-3 20:21

楼上的数字串可以有一个统一的通项公式——Table[k*Round[Sqrt[(2 n - 1)/k + 1/4]]^2 - n, {k, 19}, {n, 50}]——OEIS没有这些数字串。

补充内容 (2025-9-6 10:37):
精妙在于: "2"不能改。"1"不能少。动"1/4"就漏解。不精妙的就免了。

王守恩 · 发表于 2025-9-6 10:49:52

northwolves 发表于 2025-8-31 17:02
$a_n=2 n + 9 floor[1/2 + \sqrt[2floor[(8 + n)/9]]]^2$

丢了！——第3个数字串——我以为很简单——虽然OEIS没有——还不是那么回事——求助一下！谢谢！！！

{1, 1, 2, 3, 4, 4, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 49, 50, 51,
52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 100, ......}
A056847——重复的数是 1, 4, 9, 16, 25, ...——a(n) = n - Round[Sqrt[n]]。

{1, 1, 2, 3, 3, 4, 5, 6, 6, 7, 8, 9, 10, 10, 11, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 28, 29, 30, 31, 32, 33, 34, 35, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 45, 46, 47, 48,
49, 50, 51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 91, 92, 93, 94, 95, 96, ......}
A083920——重复的数是 1, 3, 6, 10, 15, ...——a(n) = n - Round[Sqrt[2 n]]。

{1, 1, 2, 2, 3, 3, 4, 5, 5, 6, 7, 8, 8, 9, 10, 11, 12, 13, 13, 14, 15, 16, 17, 18, 19, 20, 21, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50,
51, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, ......}
OEIS没有——重复的数是 1, 2, 3, 5, 8, 13, 21, 34, ...——a(n) = ？？？？

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[原创] 三角形数的三角形图