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数学研发论坛»论坛 › 【数学研究】 › 难题征解 › OEIS—A290743—看不懂
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楼主: 王守恩

[求助] OEIS—A290743—看不懂

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王守恩
 楼主| 发表于 2026-3-1 15:36:34 | 显示全部楼层
接楼上——绝妙的通项公式——可以改！！！——还可以改！！！！！
  1. Table[(n (n + 1) - (s*Floor[n/s] + 2 Mod[n, s]) Floor[(n + s)/s] + 2 s + 2 (n - s) Floor[Ramp[(2 s - n)/s]])/2, {s, 2, 5}, {n, 50}]——————绝妙的通项公式！！！！！
复制代码
  1. Table[(n (n + 1) - (s*Floor[n/s] + 2 Mod[n, s]) Floor[(n + s)/s] + 2Min[n, s])/2, {s, 2, 5}, {n, 50}]——————绝妙的通项公式——————可以改！！！！！
复制代码
  1. Table[Sum[Floor[(s - 1) k/s], {k, n}] + Min[n, s], {s, 2, 5}, {n, 50}]————绝妙的通项公式————可以改！！！————还可以改！！！！！
复制代码

———求具体解(具体"王牌单词”与具体“王牌碎片”)代码可以调整吗？？？？？——————谢谢各位！！！！！
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 2026-3-2 06:53:50 | 显示全部楼层
接楼上——绝妙的通项公式——可以改！！！——还可以改！！！！！——继续改！！！！！！！
  1. Table[(n (n + 1) - (s*Floor[n/s] + 2 Mod[n, s]) Floor[(n + s)/s] + 2 s + 2 (n - s) Floor[Ramp[(2 s - n)/s]])/2, {s, 2, 5}, {n, 50}]——————绝妙的通项公式！！！！！
复制代码
  1. Table[(n (n + 1) - (s*Floor[n/s] + 2 Mod[n, s]) Floor[(n + s)/s] + 2Min[n, s])/2, {s, 2, 5}, {n, 50}]——————绝妙的通项公式——————可以改！！！！！
复制代码
  1. Table[Sum[Floor[(s - 1) k/s], {k, n}] + Min[n, s], {s, 2, 5}, {n, 50}]————绝妙的通项公式————可以改！！！————还可以改！！！！！
复制代码
  1. Table[Floor[(S - 1)n^2/(2S)] + Min[n, S], {S, 2, 5}, {n, 50}]——绝妙的通项公式——可以改！——还可以改！！——继续改！！！
复制代码

———求具体解(具体"王牌单词”与具体“王牌碎片”)代码可以调整吗？？？？？——————谢谢各位！！！！！

补充内容 (2026-3-3 16:32):
第4个公式: S=2,3,4,5,6,7没问题。S=8,9,10,...有问题。
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 2026-3-3 14:14:15 | 显示全部楼层
S=2,具体解。
a(1)=1, {"0", {0}}——a(1)=1, {"0", {0}}——这样理解,后面数据出来就流畅了。
a(2)=3, {"01", {0, 1, 01}}
a(3)=4, {"001", {0, 1, 01, 001}}——"001" =“Lyndon”= “王牌单词”——{0, 1, 01, 001}=“王牌碎片”。
a(4)=6, {"0011", {0, 1, 01, 001, 011, 0011}}
a(5)=8, {"00011", {0, 1, 01, 001, 011, 0001, 0011, 00011}}
a(6)=11, {"000111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00011, 00111, 000111}}
a(7)=14, {"0000111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00111, 000011, 000111, 0000111}}
a(8)=18, {"00001111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00111, 01111, 000011, 000111, 001111, 0000111, 0001111, 00001111}}
a(9)=22, {"000001111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00111, 01111, 000001, 000011, 000111, 001111, 0000011, 0000111, 0001111, 00000111, 00001111, 000001111}}
\(\cdots\cdots\cdots\)
S=3,具体解。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=8, {"0012", {0, 1, 2, 01, 12, 001, 012, 0012}}
a(5)=11, {"00112", {0, 1, 2, 01, 12, 001, 011, 112, 0011, 0112, 00112}}
a(6)=15, {"001122", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0011, 0112, 1122, 00112, 01122, 001122}}
a(7)=19, {"0001122", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0001, 0011, 0112, 1122, 00011, 00112, 01122, 000112, 001122, 0001122}}
a(8)=24, {"00011122", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0001, 0011, 0111, 1112, 1122, 00011, 00111, 01112, 11122, 000111, 001112, 011122, 0001112, 0011122, 00011122}}
a(9)=30, {"000111222", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0001, 0011, 0111, 1112, 1122, 1222, 00011, 00111, 01112, 11122, 11222, 000111, 001112, 011122, 111222, 0001112, 0011122, 0111222, 00011122, 00111222, 000111222}}
\(\cdots\cdots\cdots\)
S=4,具体解。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=10, {"0123", {0, 1, 2, 3, 01, 12, 23, 012, 123, 0123}}
a(5)=13, {"00123", {0, 1, 2, 3, 01, 12, 23, 001, 012, 123, 0012, 0123, 00123}}
a(6)=17, {"001123", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 123, 0011, 0112, 1123, 00112, 01123, 001123}}
a(7)=22, {"0011223", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 122, 223, 0011, 0112, 1122, 1223, 00112, 01122, 11223, 001122, 011223, 0011223}}
a(8)=28, {"00112233", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 122, 223, 233, 0011, 0112, 1122, 1223, 2233, 00112, 01122, 11223, 12233, 001122, 011223, 112233, 0011223, 0112233, 00112233}}
a(9)=34, {"000112233", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 122, 223, 233, 0001, 0011, 0112, 1122, 1223, 2233, 00011, 00112, 01122, 11223, 12233, 000112, 001122, 011223, 112233, 0001122, 0011223, 0112233, 00011223, 00112233, 000112233}}
\(\cdots\cdots\cdots\)
S=5,具体解。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=10, {"0123", {0, 1, 2, 3, 01, 12, 23, 012, 123, 0123}}
a(5)=15, {"01234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 012, 123, 234, 0123, 1234, 01234}}
a(6)=19, {"001234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 012, 123, 234, 0012, 0123, 1234, 00123, 01234, 001234}}
a(7)=24, {"0011234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 011, 112, 123, 234, 0011, 0112, 1123, 1234, 00112, 01123, 11234, 001123, 011234, 0011234}}
a(8)=30, {"00112234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 011, 112, 122, 223, 234, 0011, 0112, 1122, 1223, 2234, 00112, 01122, 11223, 12234, 001122, 011223, 112234, 0011223, 0112234, 00112234}}
a(9)=37, {"001122334", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 011, 112, 122, 223, 233, 334, 0011, 0112, 1122, 1223, 2233, 2334, 00112, 01122, 11223, 12233, 22334, 001122, 011223, 112233, 122334, 0011223, 0112233, 1122334, 00112233, 01122334, 001122334}}
\(\cdots\cdots\cdots\)
\(\cdots\cdots\cdots\)
  1. Q[s_, n_] := Module[{A, B, d, T, R}, A = FrobeniusSolve[ConstantArray[1, s], n]; ——————这代码可以调整吗？？？？？
  2. B = (Total[Boole[# > 0] & /[url=home.php?mod=space&uid=6175]@[/url] #] + Sum[#[[i]]*#[[j]], {i, s - 1}, {j, i + 1, s}]) & /@ A;
  3. d = Max[B]; T = Flatten[Position[B, d]]; R = A[[T]]; If[Length[R] > 1, R = Sort[R, Function[{a, b},
  4. Module[{i = 1}, While[i <= s && a[[i]] == b[[i]], i++]; If[i > s, False, a[[i]] > b[[i]]]]]];]; First[R]]
  5. G[s_, n_] := Module[{U, V, W, f}, U = Q[s, n]; V = StringJoin[Flatten[Table[ConstantArray[ToString[i - 1], U[[i]]], {i, s}]]];
  6. W = {}; Do[Do[AppendTo[W, StringTake[V, {i, j}]], {j, i, StringLength[V]}], {i, 1, StringLength[V]}]; W = DeleteDuplicates[W];
  7. f = Select[W, StringLength[#] == 1 || Length[Union[Characters[#]]] > 1 &]; f = SortBy[f, {StringLength, # &}]; {V, f}]
  8. Do[{V, f} = G[5, n]; P = "a(" <> ToString[n] <> ")=" <> ToString[Length[f]] <> ", {"" <> V <> "", {" <> StringRiffle[f, ", "] <> "}}"; Print[P], {n, 9}]
复制代码
  1. Q[s_, n_] := First@Sort[MaximalBy[FrobeniusSolve[ConstantArray[1, s], n], (Count[#, _?Positive] + Sum[#[[i]] #[[j]], {i, s - 1}, {j, i + 1, s}]) &],
  2.    Function[{a, b}, Module[{i = 1}, While[i <= s && a[[i]] == b[[i]], i++]; i > s || a[[i]] > b[[i]]]]]
  3. G[s_, n_] := Module[{v, f}, v = StringJoin @@MapThread[ConstantArray, {ToString /@ Range[0, s - 1], Q[s, n]}];
  4.   f = Union@Flatten@Table[StringTake[v, {i, j}], {i, StringLength[v]}, {j, i, StringLength[v]}];
  5.   f = SortBy[Select[f, StringLength[#] == 1 || Length[Union@Characters@#] > 1 &], {StringLength, # &}]; {v, f}]
  6. Do[Print["a(", n, ")=", Length[#2], ", {"", #1, "", {", StringRiffle[#2, ", "], "}}"] & @@ G[5, n], {n, 9}]
复制代码
  1. Q[s_, n_] := First@MaximalBy[FrobeniusSolve[ConstantArray[1, s], n], (Count[#, _?Positive] + Sum[#[[i]] #[[j]], {i, s - 1}, {j, i + 1, s}]) &]
  2. G[s_, n_] := Module[{v, f}, v = StringJoin @@MapThread[ConstantArray, {ToString /@ Range[0, s - 1], Q[s, n]}];
  3.   f = Union@Flatten@Table[StringTake[v, {i, j}], {i, StringLength[v]}, {j, i, StringLength[v]}];
  4.   f = Select[f, StringLength[#] == 1 || Length[Union@Characters@#] > 1 &]; f = SortBy[f, {StringLength, # &}]; {v, f}]
  5. Do[Print["a(", n, ")=", Length[#2], ", {"", #1, "", {", StringRiffle[#2, ", "], "}}"] & @@ G[5, n], {n, 9}]
复制代码

3个代码功能是一样的。
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王守恩
 楼主| 发表于 2026-3-8 09:26:50 | 显示全部楼层
小结——圆满解决——OEIS—A290743
A290743——Maximum number of distinct Lyndon factors that can appear in words of length n over an alphabet of size 2.
A290743——在大小为 2 的字母表上, 长度为 n 的单词中可以出现的不同林顿因子的最大数量。
S=2,基本解。 1, 3, 4, 6, 8, 11, 14, 18, 22, 27, 32, 38, 44, 51, 58, 66, 74, 83, 92, 102, 112, 123, 134, 146, 158, 171, 184, 198, 212, 227, 242, 258, 274, 291, 308, 326, 344, 363, 382, 402, 422, 443, 464, 486, 508,

"在大小为 S 的字母表上, 长度为 n 的单词中可以出现的不同林顿因子的最大数量"问题。——问题分2块:  1,S=n,基本解。2,S=n,具体解(具体的"王牌单词”与具体的“王牌碎片”)。

S=2,基本解。{1, 3, 4, 6, 8, 11, 14, 18, 22, 27, 32, 38, 44, 51, 58, 66, 74, 83, 92, 102, 112, 123, 134, 146, 158, 171, 184, 198, 212, 227, 242, 258, 274, 291, 308, 326, 344, 363, 382, 402, 422, 443, 464, 486, 508, 531, 554, 578, 602, 627},
S=3,基本解。{1, 3, 6, 8, 11, 15, 19, 24, 30, 36, 43, 51, 59, 68, 78, 88, 99, 111, 123, 136, 150, 164, 179, 195, 211, 228, 246, 264, 283, 303, 323, 344, 366, 388, 411, 435, 459, 484, 510, 536, 563, 591, 619, 648, 678, 708, 739, 771, 803, 836},
S=4,基本解。{1, 3, 6, 10, 13, 17, 22, 28, 34, 41, 49, 58, 67, 77, 88, 100, 112, 125, 139, 154, 169, 185, 202, 220, 238, 257, 277, 298, 319, 341, 364, 388, 412, 437, 463, 490, 517, 545, 574, 604, 634, 665, 697, 730, 763, 797, 832, 868, 904, 941},
S=5,基本解。{1, 3, 6, 10, 15, 19, 24, 30, 37, 45, 53, 62, 72, 83, 95, 107, 120, 134, 149, 165, 181, 198, 216, 235, 255, 275, 296, 318, 341, 365, 389, 414, 440, 467, 495, 523, 552, 582, 613, 645, 677, 710, 744, 779, 815, 851, 888, 926, 965, 1005}}
\(\cdots\cdots\cdots\)
  1. Table[(n (n + 1) - (s*Floor[n/s] + 2 Mod[n, s]) Floor[(n + s)/s] + 2 s + 2 (n - s) Floor[Ramp[(2 s - n)/s]])/2, {s, 2, 5}, {n, 50}]
复制代码
  1. Table[(n (n + 1) - (s*Floor[n/s] + 2 Mod[n, s]) Floor[(n + s)/s] + 2Min[n, s])/2, {s, 2, 5}, {n, 50}]
复制代码
  1. Table[Sum[Floor[(s - 1) k/s], {k, n}] + Min[n, s], {s, 2, 5}, {n, 50}]
复制代码

基本解——用这3个公式都可以——OEIS——可是没有这些数字串的！！！——就是有几个——肯定没有这样痛快。

S=2,具体解。
a(1)=1, {"0", {0}}——a(1)=1, {"0", {0}}——这样理解,后面数据出来就流畅了。
a(2)=3, {"01", {0, 1, 01}}
a(3)=4, {"001", {0, 1, 01, 001}}——"001" =“Lyndon”= “王牌单词”——{0, 1, 01, 001}=“王牌碎片”。
a(4)=6, {"0011", {0, 1, 01, 001, 011, 0011}}
a(5)=8, {"00011", {0, 1, 01, 001, 011, 0001, 0011, 00011}}
a(6)=11, {"000111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00011, 00111, 000111}}
a(7)=14, {"0000111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00111, 000011, 000111, 0000111}}
a(8)=18, {"00001111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00111, 01111, 000011, 000111, 001111, 0000111, 0001111, 00001111}}
a(9)=22, {"000001111", {0, 1, 01, 001, 011, 0001, 0011, 0111, 00001, 00011, 00111, 01111, 000001, 000011, 000111, 001111, 0000011, 0000111, 0001111, 00000111, 00001111, 000001111}}
\(\cdots\cdots\cdots\)
S=3,具体解。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=8, {"0012", {0, 1, 2, 01, 12, 001, 012, 0012}}
a(5)=11, {"00112", {0, 1, 2, 01, 12, 001, 011, 112, 0011, 0112, 00112}}
a(6)=15, {"001122", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0011, 0112, 1122, 00112, 01122, 001122}}
a(7)=19, {"0001122", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0001, 0011, 0112, 1122, 00011, 00112, 01122, 000112, 001122, 0001122}}
a(8)=24, {"00011122", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0001, 0011, 0111, 1112, 1122, 00011, 00111, 01112, 11122, 000111, 001112, 011122, 0001112, 0011122, 00011122}}
a(9)=30, {"000111222", {0, 1, 2, 01, 12, 001, 011, 112, 122, 0001, 0011, 0111, 1112, 1122, 1222, 00011, 00111, 01112, 11122, 11222, 000111, 001112, 011122, 111222, 0001112, 0011122, 0111222, 00011122, 00111222, 000111222}}
\(\cdots\cdots\cdots\)
S=4,具体解。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=10, {"0123", {0, 1, 2, 3, 01, 12, 23, 012, 123, 0123}}
a(5)=13, {"00123", {0, 1, 2, 3, 01, 12, 23, 001, 012, 123, 0012, 0123, 00123}}
a(6)=17, {"001123", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 123, 0011, 0112, 1123, 00112, 01123, 001123}}
a(7)=22, {"0011223", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 122, 223, 0011, 0112, 1122, 1223, 00112, 01122, 11223, 001122, 011223, 0011223}}
a(8)=28, {"00112233", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 122, 223, 233, 0011, 0112, 1122, 1223, 2233, 00112, 01122, 11223, 12233, 001122, 011223, 112233, 0011223, 0112233, 00112233}}
a(9)=34, {"000112233", {0, 1, 2, 3, 01, 12, 23, 001, 011, 112, 122, 223, 233, 0001, 0011, 0112, 1122, 1223, 2233, 00011, 00112, 01122, 11223, 12233, 000112, 001122, 011223, 112233, 0001122, 0011223, 0112233, 00011223, 00112233, 000112233}}
\(\cdots\cdots\cdots\)
S=5,具体解。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=10, {"0123", {0, 1, 2, 3, 01, 12, 23, 012, 123, 0123}}
a(5)=15, {"01234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 012, 123, 234, 0123, 1234, 01234}}
a(6)=19, {"001234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 012, 123, 234, 0012, 0123, 1234, 00123, 01234, 001234}}
a(7)=24, {"0011234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 011, 112, 123, 234, 0011, 0112, 1123, 1234, 00112, 01123, 11234, 001123, 011234, 0011234}}
a(8)=30, {"00112234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 011, 112, 122, 223, 234, 0011, 0112, 1122, 1223, 2234, 00112, 01122, 11223, 12234, 001122, 011223, 112234, 0011223, 0112234, 00112234}}
a(9)=37, {"001122334", {0, 1, 2, 3, 4, 01, 12, 23, 34, 001, 011, 112, 122, 223, 233, 334, 0011, 0112, 1122, 1223, 2233, 2334, 00112, 01122, 11223, 12233, 22334, 001122, 011223, 112233, 122334, 0011223, 0112233, 1122334, 00112233, 01122334, 001122334}}
\(\cdots\cdots\cdots\)
\(\cdots\cdots\cdots\)
  1. Q[s_, n_] := Last@SortBy[MaximalBy[FrobeniusSolve[ConstantArray[1, s], n], (Count[#, _?Positive] + Sum[#[[i]] #[[j]], {i, s - 1}, {j, i + 1, s}]) &], # &]
  2. G[s_, n_] := Module[{v, f}, v = StringJoin @@ MapThread[ConstantArray, {ToString /@ Range[0, s - 1], Q[s, n]}];
  3.   f = Union@Flatten@Table[StringTake[v, {i, j}], {i, StringLength[v]}, {j, i, StringLength[v]}];
  4.   f = Select[f, StringLength[#] == 1 || Length[Union@Characters@#] > 1 &]; f = SortBy[f, {StringLength, # &}]; {v, f}]
  5. Do[{v, f} = G[5, n]; Print["a(", n, ")=", Length[f], ", {"", v, "", {", StringRiffle[f, ", "], "}}"], {n, 9}]
复制代码

特别说明——主帖与“彩珠手串的配色计数”很是有关联。https://bbs.emath.ac.cn/forum.ph ... 61&fromuid=9898(出处: 数学研发论坛)——具体解是其中一个不复杂的彩珠手串。

{1, 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},
{1, 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},
{1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310},
{1, 8, 39, 136, 377, 888, 1855, 3536, 6273, 10504, 16775, 25752, 38233, 55160, 77631, 106912, 144449, 191880, 251047, 324008, 413049, 520696, 649727, 803184, 984385, 1196936, 1444743, 1732024, 2063321, 2443512, 2877823, 3371840}
\(\cdots\cdots\cdots\)
  1. Table[Total[EulerPhi[n/#] m^# & /@ Divisors[n]/(2 n)] + (m^Ceiling[n/2] + m^Ceiling[(n + 1)/2])/4, {n, 2, 5}, {m, 34}]
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王守恩
 楼主| 发表于 2026-3-13 10:23:02 | 显示全部楼层
接楼上。   m 种颜色 n 颗珠。
{{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},
{1, 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},
{1, 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},
{1, 6, 21, 55, 120, 231, 406, 666, 1035, 1540, 2211, 3081, 4186, 5565, 7260, 9316, 11781, 14706, 18145, 22155, 26796, 32131, 38226, 45150, 52975, 61776, 71631, 82621, 94830, 108345, 123256, 139656, 157641, 177310},
{1, 8, 39, 136, 377, 888, 1855, 3536, 6273, 10504, 16775, 25752, 38233, 55160, 77631, 106912, 144449, 191880, 251047, 324008, 413049, 520696, 649727, 803184, 984385, 1196936, 1444743, 1732024, 2063321, 2443512, 2877823,
{1, 13, 92, 430, 1505, 4291, 10528, 23052, 46185, 86185, 151756, 254618, 410137, 638015, 963040, 1415896, 2034033, 2862597, 3955420, 5376070, 7198961, 9510523, 12410432, 16012900, 20448025, 25863201, 32424588, 40318642,
{1, 18, 198, 1300, 5895, 20646, 60028, 151848, 344925, 719290, 1399266, 2569788, 4496323, 7548750, 12229560, 19206736, 29351673, 43782498, 63913150, 91508580, 128746431, 178285558, 243341748, 327771000, 436160725,
{1, 30, 498, 4435, 25395, 107331, 365260, 1058058, 2707245, 6278140, 13442286, 26942565, 51084943, 92383305, 160386360, 268718116, 436365945, 689252778, 1062132490, 1600850055, 2365010571, 3431103775, 4896133188,
{1, 46, 1219, 15084, 110085, 563786, 2250311, 7472984, 21552969, 55605670, 131077771, 286779076, 589324749, 1148105154, 2136122255, 3818273456, 6588925841, 11020906014, 17928333139, 28446045340, 44128712341,  
Table[DivisorSum[n,EulerPhi[n/#]m^# &]/(2n)+(m^Ceiling[n/2]+m^Ceiling[(n+1)/2])/4,{n,9},{m,34}]——注意这个公式与楼上公式的不同之处。

不分颜色 n 颗珠。——是这样一串数。
{1, 3, 10, 55, 377, 4291, 60028, 1058058, 21552969, 500280022, 12969598086, 371514016094, 11649073935505, 396857785692525, 14596464294191704, 576460770691256356, 24330595997127372497, 1092955780817066765469,
Table[DivisorSum[n, EulerPhi[#] n^(n/#) &]/(2 n) + (n^Ceiling[n/2] + n^Ceiling[(n + 1)/2])/4, {n, 20}]——注意这个公式与楼上公式的相同之处。
Table[If[n == 2, 3, CycleIndexPolynomial[DihedralGroup[n], Array[s, n]] /. s[_] -> n], {n, 20}]——这个也可以！！

n 种颜色 n 颗珠。——是这样一串数。
{1, 1, 3, 12, 60, 360, 2520, 20160, 181440, 1814400, 19958400, 239500800, 3113510400, 43589145600, 653837184000, 10461394944000, 177843714048000, 3201186852864000, 60822550204416000, 1216451004088320000}
Table[GroupOrder[AlternatingGroup[n]], {n, 20}]——当然, 可以用楼上的公式！！！

"在大小为 S 的字母表上, 长度为 n 的单词中可以出现的不同林顿因子的最大数量"问题。——问题分2块:  1,S=n,基本解。
{1, 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, 990, 1035, 1081, 1128, 1176, 1225, 1275}

"在大小为 S 的字母表上, 长度为 n 的单词中可以出现的不同林顿因子的最大数量"问题。——问题分2块:  2,S=n,具体解(具体的"王牌单词”与具体的“王牌碎片”)。
a(1)=1, {"0", {0}}
a(2)=3, {"01", {0, 1, 01}}
a(3)=6, {"012", {0, 1, 2, 01, 12, 012}}
a(4)=10, {"0123", {0, 1, 2, 3, 01, 12, 23, 012, 123, 0123}}
a(5)=15, {"01234", {0, 1, 2, 3, 4, 01, 12, 23, 34, 012, 123, 234, 0123, 1234, 01234}}
a(6)=21, {"012345", {0, 1, 2, 3, 4, 5, 01, 12, 23, 34, 45, 012, 123, 234, 345, 0123, 1234, 2345, 01234, 12345, 012345}}
a(7)=28, {"0123456", {0, 1, 2, 3, 4, 5, 6, 01, 12, 23, 34, 45, 56, 012, 123, 234, 345, 456, 0123, 1234, 2345, 3456, 01234, 12345, 23456, 012345, 123456, 0123456}}
a(8)=36, {"01234567", {0, 1, 2, 3, 4, 5, 6, 7, 01, 12, 23, 34, 45, 56, 67, 012, 123, 234, 345, 456, 567, 0123, 1234, 2345, 3456, 4567, 01234, 12345, 23456, 34567, 012345, 123456, 234567, 0123456, 1234567, 01234567}}
a(9)=45, {"012345678", {0, 1, 2, 3, 4, 5, 6, 7, 8, 01, 12, 23, 34, 45, 56, 67, 78, 012, 123, 234, 345, 456, 567, 678, 0123, 1234, 2345, 3456, 4567, 5678, 01234, 12345, 23456, 34567, 45678, 012345, 123456, 234567, 345678, 0123456, 1234567, 2345678, 01234567, 12345678, 012345678}}
a(10)=55, {"0123456789", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 01, 12, 23, 34, 45, 56, 67, 78, 89, 012, 123, 234, 345, 456, 567, 678, 789, 0123, 1234, 2345, 3456, 4567, 5678, 6789, 01234, 12345, 23456, 34567, 45678, 56789, 012345, 123456, 234567,
345678, 456789, 0123456, 1234567, 2345678, 3456789, 01234567, 12345678, 23456789, 012345678, 123456789, 0123456789}}
a(11)=66, {"0123456789A", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, 012, 123, 234, 345, 456, 567, 678, 789, 89A, 0123, 1234, 2345, 3456, 4567, 5678, 6789, 789A, 01234, 12345, 23456, 34567, 45678, 56789, 6789A,
012345, 123456, 234567, 345678, 456789, 56789A, 0123456, 1234567, 2345678, 3456789, 456789A, 01234567, 12345678, 23456789, 3456789A, 012345678, 123456789, 23456789A, 0123456789, 123456789A, 0123456789A}}
a(12)=78, {"0123456789AB", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, AB, 012, 123, 234, 345, 456, 567, 678, 789, 89A, 9AB, 0123, 1234, 2345, 3456, 4567, 5678, 6789, 789A, 89AB, 01234, 12345, 23456, 34567, 45678,
56789, 6789A, 789AB, 012345, 123456, 234567, 345678, 456789, 56789A, 6789AB, 0123456, 1234567, 2345678, 3456789, 456789A, 56789AB, 01234567, 12345678, 23456789, 3456789A, 456789AB, 012345678, 123456789, 23456789A, 3456789AB,
0123456789, 123456789A, 23456789AB, 0123456789A, 123456789AB, 0123456789AB}}
a(13)=91, {"0123456789ABC", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, AB, BC, 012, 123, 234, 345, 456, 567, 678, 789, 89A, 9AB, ABC, 0123, 1234, 2345, 3456, 4567, 5678, 6789, 789A, 89AB, 9ABC, 01234, 12345,
23456, 34567, 45678, 56789, 6789A, 789AB, 89ABC, 012345, 123456, 234567, 345678, 456789, 56789A, 6789AB, 789ABC, 0123456, 1234567, 2345678, 3456789, 456789A, 56789AB, 6789ABC, 01234567, 12345678, 23456789, 3456789A, 456789AB,
56789ABC, 012345678, 123456789, 23456789A, 3456789AB, 456789ABC, 0123456789, 123456789A, 23456789AB, 3456789ABC, 0123456789A, 123456789AB, 23456789ABC, 0123456789AB, 123456789ABC, 0123456789ABC}}
a(14)=105, {"0123456789ABCD", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, AB, BC, CD, 012, 123, 234, 345, 456, 567, 678, 789, 89A, 9AB, ABC, BCD, 0123, 1234, 2345, 3456, 4567, 5678, 6789, 789A, 89AB, 9ABC,
ABCD, 01234, 12345, 23456, 34567, 45678, 56789, 6789A, 789AB, 89ABC, 9ABCD, 012345, 123456, 234567, 345678, 456789, 56789A, 6789AB, 789ABC, 89ABCD, 0123456, 1234567, 2345678, 3456789, 456789A, 56789AB, 6789ABC, 789ABCD,
01234567, 12345678, 23456789,3456789A, 456789AB, 56789ABC, 6789ABCD, 012345678, 123456789, 23456789A, 3456789AB, 456789ABC, 56789ABCD, 0123456789, 123456789A, 23456789AB, 3456789ABC, 456789ABCD, 0123456789A,
123456789AB, 23456789ABC, 3456789ABCD, 0123456789AB, 123456789ABC, 23456789ABCD, 0123456789ABC, 123456789ABCD, 0123456789ABCD}}
a(15)=120, {"0123456789ABCDE", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, AB, BC, CD, DE, 012, 123, 234, 345, 456, 567, 678, 789, 89A, 9AB, ABC, BCD, CDE, 0123, 1234, 2345, 3456, 4567, 5678, 6789, 789A,
89AB, 9ABC, ABCD, BCDE, 01234, 12345, 23456, 34567, 45678, 56789, 6789A, 789AB, 89ABC, 9ABCD, ABCDE, 012345, 123456, 234567, 345678, 456789, 56789A, 6789AB, 789ABC, 89ABCD, 9ABCDE, 0123456, 1234567, 2345678, 3456789, 456789A,
56789AB, 6789ABC, 789ABCD, 89ABCDE, 01234567,12345678, 23456789, 3456789A, 456789AB, 56789ABC, 6789ABCD, 789ABCDE, 012345678, 123456789, 23456789A, 3456789AB, 456789ABC, 56789ABCD, 6789ABCDE, 0123456789, 123456789A,
23456789AB, 3456789ABC, 456789ABCD, 56789ABCDE, 0123456789A, 123456789AB, 23456789ABC, 3456789ABCD, 456789ABCDE, 0123456789AB, 123456789ABC, 23456789ABCD, 3456789ABCDE, 0123456789ABC, 123456789ABCD,
23456789ABCDE, 0123456789ABCD, 123456789ABCDE, 0123456789ABCDE}}
a(16)=136, {"0123456789ABCDEF", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, AB, BC, CD, DE, EF, 012, 123, 234, 345, 456, 567, 678, 789, 89A, 9AB, ABC, BCD, CDE, DEF, 0123, 1234, 2345, 3456, 4567, 5678,
6789, 789A, 89AB, 9ABC, ABCD, BCDE, CDEF, 01234, 12345, 23456, 34567, 45678, 56789, 6789A, 789AB, 89ABC, 9ABCD, ABCDE, BCDEF, 012345, 123456, 234567, 345678, 456789, 56789A, 6789AB, 789ABC, 89ABCD, 9ABCDE, ABCDEF, 0123456,
1234567, 2345678, 3456789, 456789A, 56789AB, 6789ABC, 789ABCD, 89ABCDE, 9ABCDEF, 01234567, 12345678, 23456789, 3456789A, 456789AB, 56789ABC, 6789ABCD, 789ABCDE, 89ABCDEF, 012345678, 123456789, 23456789A, 3456789AB,
456789ABC, 56789ABCD, 6789ABCDE, 789ABCDEF, 0123456789, 123456789A, 23456789AB, 3456789ABC, 456789ABCD, 56789ABCDE, 6789ABCDEF, 0123456789A, 123456789AB, 23456789ABC, 3456789ABCD, 456789ABCDE, 56789ABCDEF,
0123456789AB, 123456789ABC, 23456789ABCD, 3456789ABCDE, 456789ABCDEF, 0123456789ABC, 123456789ABCD, 23456789ABCDE, 3456789ABCDEF, 0123456789ABCD, 123456789ABCDE, 23456789ABCDEF, 0123456789ABCDE, 123456789ABCDEF, 0123456789ABCDEF}}
  1. (*无限扩展的字符映射：0-9数字，10-35大写，36-61小写，62-...可继续添加*)
  2. t[i_] := Which[i < 10, ToString[i], i < 36, FromCharacterCode[65 + i - 10],i < 62,  FromCharacterCode[97 + i - 36],True, "?"(*此处可继续添加，例如希腊字母：i<88,FromCharacterCode[944+i-62] 等*)]
  3. g[n_] := Module[{v, s}, v = StringJoin[t /@ Range[0, n - 1]];  s = Flatten[Table[StringTake[v, {i, j}], {i, n}, {j, i, n}]];  s = SortBy[s, {StringLength, # &}]; {v, s}]
  4. Print[StringJoin[Table[ToString[StringForm["a(``)=``, {"``", {``}}", k, Length[#2], #1, StringRiffle[#2, ", "]] & @@ g[k]] <> "\n", {k, 9}]]]
复制代码

各位——这代码复制了我自己也不能用——什么原因？——原代码在我的Mathematica12.1肯定是可以用的！

(*无限扩展的字符映射：0-9数字，10-35大写，36-61小写，62-...可继续添加*)
t[i_] := Which[i < 10, ToString, i < 36, FromCharacterCode[65 + i - 10], i < 62, FromCharacterCode[97 + i - 36], True, "?"(*此处可继续添加，例如希腊字母：i<88, FromCharacterCode[944+i-62] 等*)]
g[n_] := Module[{v, s}, v = StringJoin[t /@ Range[0, n - 1]]; s = Flatten[Table[StringTake[v, {i, j}], {i, n}, {j, i, n}]]; s = SortBy[s, {StringLength, # &}]; {v, s}]
Print[StringJoin[Table[ToString[StringForm["a(``)=``, {\"``\", {``}}", k, Length[#2], #1, StringRiffle[#2, ", "]] & @@ g[k]] <> "\n", {k, 9}]]]
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 2026-3-14 16:19:17 | 显示全部楼层
本帖最后由 王守恩 于 2026-3-14 16:31 编辑

G[s_, n_] := Module[{q = Quotient[n, s], r = Mod[n, s], v, u, w}, v = StringJoin @@MapThread[ConstantArray, {(If[# < 10, ToString@#, FromCharacterCode[# + 55]] &) /@ Range[0, s - 1], Join[ConstantArray[q + 1, r], ConstantArray[q, s - r]]}];
w = StringLength[v]; u = Union[Flatten[Table[StringTake[v, {i, j}], {i, w}, {j, i, w}]]]; {v, SortBy[Select[u, StringLength[#] == 1 || Length[Union@Characters@#] > 1 &], {StringLength, # &}]}]
Print[StringForm["a(19)=``, {\"``\", {``}}", Length[#2], #1, StringRiffle[#2, ", "]] & @@ G[13, 19]]——其中:   "a(19)=`````` , {\"``\", {``}}",  有问题。 Mathematica 中，StringForm 使用双反引号``作为占位符\(\acute\ldotp\grave\grave\)
a(19)=178, {"0011223344556789ABC", {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, 01, 12, 23, 34, 45, 56, 67, 78, 89, 9A, AB, BC, 001, 011, 112, 122, 223, 233, 334, 344, 445, 455, 556, 567, 678, 789, 89A, 9AB, ABC, 0011, 0112, 1122, 1223, 2233, 2334, 3344, 3445, 4455, 4556, 5567, 5678, 6789, 789A, 89AB, 9ABC, 00112, 01122, 11223, 12233, 22334, 23344, 33445, 34455, 44556, 45567, 55678, 56789, 6789A, 789AB, 89ABC, 001122, 011223, 112233, 122334, 223344, 233445, 334455, 344556, 445567, 455678, 556789, 56789A, 6789AB, 789ABC, 0011223, 0112233, 1122334, 1223344, 2233445, 2334455, 3344556, 3445567, 4455678, 4556789, 556789A, 56789AB, 6789ABC, 00112233, 01122334, 11223344, 12233445, 22334455, 23344556, 33445567, 34455678, 44556789, 4556789A, 556789AB, 56789ABC, 001122334, 011223344, 112233445, 122334455, 223344556, 233445567, 334455678, 344556789, 44556789A, 4556789AB, 556789ABC, 0011223344, 0112233445, 1122334455, 1223344556, 2233445567, 2334455678, 3344556789, 344556789A, 44556789AB, 4556789ABC, 00112233445, 01122334455, 11223344556, 12233445567, 22334455678, 23344556789, 3344556789A, 344556789AB, 44556789ABC, 001122334455, 011223344556, 112233445567, 122334455678, 223344556789, 23344556789A, 3344556789AB, 344556789ABC, 0011223344556, 0112233445567, 1122334455678, 1223344556789, 223344556789A, 23344556789AB, 3344556789ABC, 00112233445567, 01122334455678, 11223344556789, 1223344556789A, 223344556789AB, 23344556789ABC, 001122334455678, 011223344556789, 11223344556789A, 1223344556789AB, 223344556789ABC, 0011223344556789, 011223344556789A, 11223344556789AB, 1223344556789ABC, 0011223344556789A, 011223344556789AB, 11223344556789ABC, 0011223344556789AB, 011223344556789ABC, 0011223344556789ABC}}

Table[Sum[Floor[(s - 1) k/s], {k, n}] + Min[n, s], {s, 13, 13}, {n, 19, 19}]——{{178}}——a(19)=178的佐证。
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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