三角形数的三角形图

王守恩

A062708——有个三角形数的三角形图(网页上找不到类似的图)——你来画得比它大一些(我不会画)——我来出道题——蛮好玩的！谢谢！

题目是这样。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,
a(17)=70,
a(18)=72,
a(19)=74,
a(20)=76,
a(21)=78,
a(22)=80,
a(23)=82,
a(24)=84,
a(25)=86,
a(26)=88,
a(27)=90,
a(28)=137,
a(29)=139,
a(30)=141,
a(31)=143,
a(32)=145,
......
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, 143, 145,......,求通项公式。

王守恩

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, 143, 145,......,求通项公式。——这通项公式真还不好找(没头绪)。求助各位。

我们先往前走。记"0"关于"n"的对称数为"a(n)"。如果把 "0", "n", "a(n)" 这条直线拉出来。譬如
b(1)={0, 1, 11, 30, 58, 95, 141, 196, 260, 333, 415, 506, 606, 715, 833, 960, 1096, 1241, 1395, 1558, 1730, 1911, 2101, 2300, 2508, 2725, 2951, 3186, 3430, 3683}
b(2)={0, 2, 13, 33, 62, 100, 147, 203, 268, 342, 425, 517, 618, 728, 847, 975, 1112, 1258, 1413, 1577, 1750, 1932, 2123, 2323, 2532, 2750, 2977, 3213, 3458, 3712}
b(3)={0, 3, 15, 36, 66, 105, 153, 210, 276, 351, 435, 528, 630, 741, 861, 990, 1128, 1275, 1431, 1596, 1770, 1953, 2145, 2346, 2556, 2775, 3003, 3240, 3486, 3741}
b(4)={0, 4, 17, 39, 70, 110, 159, 217, 284, 360, 445, 539, 642, 754, 875, 1005, 1144, 1292, 1449, 1615, 1790, 1974, 2167, 2369, 2580, 2800, 3029, 3267, 3514, 3770}
b(5)={0, 5, 19, 42, 74, 115, 165, 224, 292, 369, 455, 550, 654, 767, 889, 1020, 1160, 1309, 1467, 1634, 1810, 1995, 2189, 2392, 2604, 2825, 3055, 3294, 3542, 3799}
b(6)={0, 6, 21, 45, 78, 120, 171, 231, 300, 378, 465, 561, 666, 780, 903, 1035, 1176, 1326, 1485, 1653, 1830, 2016, 2211, 2415, 2628, 2850, 3081, 3321, 3570, 3828}
b(7)={0, 7, 23, 48, 82, 125, 177, 238, 308, 387, 475, 572, 678, 793, 917, 1050, 1192, 1343, 1503, 1672, 1850, 2037, 2233, 2438, 2652, 2875, 3107, 3348, 3598, 3857}
b(8)={0, 8, 25, 51, 86, 130, 183, 245, 316, 396, 485, 583, 690, 806, 931, 1065, 1208, 1360, 1521, 1691, 1870, 2058, 2255, 2461, 2676, 2900, 3133, 3375, 3626, 3886}
b(9)={0, 9, 27, 54, 90, 135, 189, 252, 324, 405, 495, 594, 702, 819, 945, 1080, 1224, 1377, 1539, 1710, 1890, 2079, 2277, 2484, 2700, 2925, 3159, 3402, 3654, 3915}
b(10)={0, 10, 56, 138, 256, 410, 600, 826, 1088, 1386, 1720, 2090, 2496, 2938, 3416, 3930, 4480, 5066, 5688, 6346, 7040, 7770, 8536, 9338, 10176, 11050, 11960, 12906, 13888, 14906}
b(11)={0, 11, 58, 141, 260, 415, 606, 833, 1096, 1395, 1730, 2101, 2508, 2951, 3430, 3945, 4496, 5083, 5706, 6365, 7060, 7791, 8558, 9361, 10200, 11075, 11986, 12933, 13916, 14935}
b(12)={0, 12, 60, 144, 264, 420, 612, 840, 1104, 1404, 1740, 2112, 2520, 2964, 3444, 3960, 4512, 5100, 5724, 6384, 7080, 7812, 8580, 9384, 10224, 11100, 12012, 12960, 13944, 14964}
b(13)={0, 13, 62, 147, 268, 425, 618, 847, 1112, 1413, 1750, 2123, 2532, 2977, 3458, 3975, 4528, 5117, 5742, 6403, 7100, 7833, 8602, 9407, 10248, 11125, 12038, 12987, 13972, 14993}
b(14)={0, 14, 64, 150, 272, 430, 624, 854, 1120, 1422, 1760, 2134, 2544, 2990, 3472, 3990, 4544, 5134, 5760, 6422, 7120, 7854, 8624, 9430, 10272, 11150, 12064, 13014, 14000, 15022}
......
b(23)={0, 23, 82, 177, 308, 475, 678, 917, 1192, 1503, 1850, 2233, 2652, 3107, 3598, 4125, 4688, 5287, 5922, 6593, 7300, 8043, 8822, 9637, 10488, 11375, 12298, 13257, 14252, 15283}
b(24)={0, 24, 84, 180, 312, 480, 684, 924, 1200, 1512, 1860, 2244, 2664, 3120, 3612, 4140, 4704, 5304, 5940, 6612, 7320, 8064, 8844, 9660, 10512, 11400, 12324, 13284, 14280, 15312}
b(25)={0, 25, 86, 183, 316, 485, 690, 931, 1208, 1521, 1870, 2255, 2676, 3133, 3626, 4155, 4720, 5321, 5958, 6631, 7340, 8085, 8866, 9683, 10536, 11425, 12350, 13311, 14308, 15341}
b(26)={0, 26, 88, 186, 320, 490, 696, 938, 1216, 1530, 1880, 2266, 2688, 3146, 3640, 4170, 4736, 5338, 5976, 6650, 7360, 8106, 8888, 9706, 10560, 11450, 12376, 13338, 14336, 15370}
b(27)={0, 27, 90, 189, 324, 495, 702, 945, 1224, 1539, 1890, 2277, 2700, 3159, 3654, 4185, 4752, 5355, 5994, 6669, 7380, 8127, 8910, 9729, 10584, 11475, 12402, 13365, 14364, 15399}
b(28)={0, 28, 137, 327, 598, 950, 1383, 1897, 2492, 3168, 3925, 4763, 5682, 6682, 7763, 8925, 10168, 11492, 12897, 14383, 15950, 17598, 19327, 21137, 23028, 25000, 27053, 29187, 31402, 33698}
b(29)={0, 29, 139, 330, 602, 955, 1389, 1904, 2500, 3177, 3935, 4774, 5694, 6695, 7777, 8940, 10184, 11509, 12915, 14402, 15970, 17619, 19349, 21160, 23052, 25025, 27079, 29214, 31430, 33727}
b(30)={0, 30, 141, 333, 606, 960, 1395, 1911, 2508, 3186, 3945, 4785, 5706, 6708, 7791, 8955, 10200, 11526, 12933, 14421, 15990, 17640, 19371, 21183, 23076, 25050, 27105, 29241, 31458, 33756}
b(31)={0, 31, 143, 336, 610, 965, 1401, 1918, 2516, 3195, 3955, 4796, 5718, 6721, 7805, 8970, 10216, 11543, 12951, 14440, 16010, 17661, 19393, 21206, 23100, 25075, 27131, 29268, 31486, 33785}
b(32)={0, 32, 145, 339, 614, 970, 1407, 1925, 2524, 3204, 3965, 4807, 5730, 6734, 7819, 8985, 10232, 11560, 12969, 14459, 16030, 17682, 19415, 21229, 23124, 25100, 27157, 29295, 31514, 33814}
......
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！

王守恩

这串数规律不好找——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, 143, 145,......,求通项公式。——恭候大神出现！

我们看看另外一串数。看能不能找个突破口。
b(1) = {8, 0, 1, 11}
b(2) = {7, 0, 2, 13}
b(3) = {6, 0, 3, 15}——用后面3个数能把这条直线拉出来。
b(4) = {5, 0, 4, 17}——用前面3个数也是能把这条直线拉出来的。
b(5) = {4, 0, 5, 19}
b(6) = {3, 0, 6, 21}
b(7) = {2, 0, 7, 23}
b(8) = {1, 0, 8, 25}
b(9) = {0, 0, 9, 27}
b(10)={26, 0, 10, 56}
b(11)={25, 0, 11, 58}
b(12)={24, 0, 12, 60}
b(13)={23, 0, 13, 62}
b(14)={22, 0, 14, 64}
......
b(23)={13, 0, 23, 82}
b(24)={12, 0, 24, 84}
b(25)={11, 0, 25, 86}
b(26)={10, 0, 26, 88}
b(27)={09, 0, 27, 90}
b(28)={53, 0, 28, 137}
b(29)={52, 0, 29, 139}
b(30)={51, 0, 30, 141}
b(31)={50, 0, 31, 143}
b(32)={49, 0, 32, 145}
......
第1个数是这样一串数。——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。
{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, 32, 31, 30, 29, 28, 27,
89, 88, 87, 86, 85, 84, 83, 82, 81, 80, 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,
134, 133, 132, 131, 130, 129, 128, 127, 126, 125, 124, 123, 122, 121, 120, 119, 118, 117, 116, 115, 114, ..., 102, 101, 100, 99, 98, 97, 96, 95, 94, 93, 92, 91, 90,
188, 187, 186, 185, 184, 183, 182, 181, 180, 179, 178, 177, 176, 175, 174, 173, 172, 171, 170, 169, 168, ..., 148, 147, 146, 145, 144, 143, 142, 141, 140, 139, 138, 137, 136, 135,
251, 250, 249, 248, 247, 246, 245, 244, 243, 242, 241, 240, 239, 238, 237, 236, 235, 234, 233, 232, 231, ...,  205, 204, 203, 202, 201, 200, 199, 198, 197, 196, 195, 194, 193, 192, 191, 190, 189,
323, 322, 321, 320, 319, 318, 317, 316, 315, 314, 313, 312, 311, 310, 309, 308, 307, 306, 305, 304, 303, ...,  271, 270, 269, 268, 267, 266, 265, 264, 263, 262, 261, 260, 259, 258, 257, 256, 255, 254, 253, 252,
404, 403, 402, 401, 400, 399, 398, 397, 396, 395, 394, 393, 392, 391, 390, 389, 388, 387, 386, 385, 384, ...,  346, 345, 344, 343, 342, 341, 340, 339, 338, 337, 336, 335, 334, 333, 332, 331, 330, 329, 328, 327, 326, 325, 324,
......

补充内容 (2025-9-4 08:27):
b(n) = Table[- n + 9 Round[Sqrt[2 (n + 1)]/3]^2, {n, 60}]

northwolves

$a_n=2 n + 9 floor[1/2 + \sqrt[2floor[(8 + n)/9]]]^2$

王守恩

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

趁热打铁！——类似的题目——记"0"关于"n"的对称数为"a(n)"。——A001107——
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,

王守恩

王守恩 发表于 2025-8-31 18:03
趁热打铁！——类似的题目——记"0"关于"n"的对称数为"a(n)"。——A001107——
a(1)=10,
a(2)=12,

{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, 134, 136, 138, 140, 142, 144, 146, 148, 150, 152, 154, 156, 158, 160, 162, 164, 166, 168}

代码1——Table[8 Floor[Sqrt[2 Floor[(n + 7)/8]] + 1/2]^2 + 2 n, {n, 200}]

代码2——Table[8 Round[Sqrt[2 Floor[(n + 7)/8]]]^2 + 2 n, {n, 200}]——代码2 = 代码1。

代码3——Flatten@Table[8 v^2 + 2 u, {v, 1, 6}, {u, (2 v - 1)^2, (2 v + 1)^2 - 1}]——代码3是 "怎么" 变到代码1的？

代码4——Table[v = Which[n < 9, 1, n < 25, 2, n < 49, 3, n < 81, 4, n < 121, 5, n < 169, 6, True, Round[Sqrt[n/2] + 1]]; 8 v^2 + 2 n, {n, 168}]——要一个一个一个一个写出来？

王守恩

用这个——可以画1楼的图——我就是不知道怎么让图显示在这里。

1. n = 13; f = {{0, 0}}; v = {1, 0}; For[i = 1, i <= n, i++, w = Last[f] + i*v; AppendTo[f, w]; v = RotationTransform[2 \[Pi]/3][v];]; p = Line[f]; r = Table[i*(i - 1)/2, {i, 1, n + 1}];
   
2. 
   
3. a = f; b = Table[Text[Style[ToString[r[[i]]], 12, Bold], f[[i]], {0, 1.5}], {i, n + 1}]; For[i = 1, i <= n, i++, s = r[[i]]; m = r[[i + 1]]; c = f[[i]]; e = f[[i + 1]]; d = e - c;
   
4. 
   
5. l = Norm[d]; u = d/l; For[j = s + 1, j < m, j++, t = c + (j - s)/(m - s)*l*u; AppendTo[a, t]; AppendTo[b, Text[Style[ToString[j], 12, Bold], t, {0, 1.5}]];];];
   
6. 
   
7. Graphics[{{Blue, Thickness[0.001], p}, {Black, PointSize[0.008], Point[a]}, {Black, b}}, AspectRatio -> Automatic, PlotRange -> All, ImageSize -> 500]

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northwolves

王守恩

northwolves 发表于 2025-9-1 14:41

这个图也可以出来？——这串数也可以出来？——好玩！！！

1. n = 9; s = {}; a = {}; AppendTo[s, {0, 0}]; AppendTo[a, Text[Style["0", 12, Bold], {0, 0}, {0, 1.5}]]; For[l = 1, l <= n, l++, r = (2*l - 1)^2; c = r; For[x = 0, x <= l, x++, p = {x, l};
   
2. 
   
3. AppendTo[s, p]; AppendTo[a, Text[Style[ToString[c], 12, Bold], p, {0, 1.5}]]; c++;]; For[y = l - 1, y >= -l, y--, p = {l, y};
   
4. 
   
5. AppendTo[s, p]; AppendTo[a, Text[Style[ToString[c], 12, Bold], p, {0, 1.5}]]; c++;]; For[x = l - 1, x >= -l, x--, p = {x, -l};
   
6. 
   
7. AppendTo[s, p]; AppendTo[a, Text[Style[ToString[c], 12, Bold], p, {0, 1.5}]]; c++;]; For[y = -l + 1, y < l, y++, p = {-l, y};
   
8. 
   
9. AppendTo[s, p]; AppendTo[a, Text[Style[ToString[c], 12, Bold], p, {0, 1.5}]]; c++;]; For[x = -l, x < 0, x++, p = {x, l};
   
10. 
    
11. AppendTo[s, p]; AppendTo[a, Text[Style[ToString[c], 12, Bold], p, {0, 1.5}]]; c++;];]; f = {}; For[l = 1, l <= n, l++, d = {};
    
12. 
    
13. For[x = -l, x <= l, x++, AppendTo[d, {x, l}]]; For[y = l - 1, y >= -l, y--, AppendTo[d, {l, y}]]; For[x = l - 1, x >= -l, x--, AppendTo[d, {x, -l}]];
    
14. 
    
15. For[y = -l + 1, y <= l; y++, AppendTo[d, {-l, y}]]; AppendTo[d, {-l, l}]; AppendTo[f, Line[d]];];
    
16. 
    
17. Graphics[{{Blue, Thickness[0.002], f}, {PointSize[0.01], Point[s]}, {a}}, AspectRatio -> Automatic, PlotRange -> All, ImageSize -> 600]

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王守恩

好玩！！！——类似的题目。

1. f[r_, s_, B_] := Module[{p, a}, p = Table[r + s*{Cos[z], Sin[z]}, {z, Table[i*2*Pi/5 + Pi/2, {i, 0, 4}]}];
   
2. 
   
3. If[B > 1, a = {}; For[i = 1, i <= 5, i++, y = If[i < 5, i + 1, 1]; For[j = 0, j < B, j++, t = j/B; G = (1 - t)*p[[i]] + t*p[[y]]; AppendTo[a, G];]]; a, p]]
   
4. 
   
5. h = 6; R = {1, 2, 3, 4, 5, 6}; k = {1, 3, 5, 7, 9, 11, 13}/(2 Sin[Pi/5]); a = {}; b = {}; d = {}; c = 0;
   
6. 
   
7. AppendTo[a, {0, 0}]; AppendTo[b, Text[Style[0, 12, Bold], {0, 0}]]; For[w = 1, w ≤ h, w++, u = f[{0, 0}, k[[w]], R[[w]]];
   
8. 
   
9. For[i = 1, i ≤ Length[u], i++, c++; AppendTo[a, u[[i]]]; V = ArcTan @@ u[[i]] + Pi/2; AppendTo[b, Text[Style[c, 10, Bold], u[[i]] + {Cos[V], Sin[V]}/5]];];
   
10. 
    
11. v = f[{0, 0}, k[[w]], 1]; For[i = 1, i ≤ 5, i++, x = If[i < 5, i + 1, 1]; AppendTo[d, {Thickness[0.00001], Blue, Line[{v[[i]], v[[x]]}]}];];];
    
12. 
    
13. Graphics[{d, {PointSize[0.01], Point[a]}, b}, AspectRatio -> Automatic, PlotRange -> All, ImageSize -> 400]

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