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楼主: 王守恩

[原创] 数字串的通项公式

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 楼主| 发表于 2025-8-12 17:56:01 | 显示全部楼层
这些数字串可都是OEIS没有的。

a(n)=  n^2——{1, 4, 9, 13, 17, 23, 29, 37, 45, 55, 65, 77, 89, 102, 111, 122, 133, 144, 157, 170, 183, 198, 213, 228, 245, 262, 279, 298, 317, 336, 357, 378, 399, 422, 445, 468, 493, 518, 543, 570, 597, 624, 653, 682,——
a(n)=10^n——{1, 10, 55, 370, 2777, 22222, 185185, 1587301, 13888888, 123456790, 1111111111, 10101010100, 91919191919, 842592592592, 7777777777777, 72222222222222, 674074074074074, 6319444444444444,——
a(n)= n^n——{1, 4, 18, 122, 1058, 11553, 155775, 2555475, 49816449, 1111111111, 26947525611, 752267629947, 22427586978811, 748207862444608, 26412058911292381, 1029360799201087511, 41917568649872393764,——
a(n)=n(n+1)/2——{1, 3, 6, 10, 12, 15, 19, 23, 27, 32, 38, 44, 50, 57, 65, 73, 81, 90, 100, 106, 113, 121, 128, 136, 145, 153, 162, 172, 181, 191, 202, 212, 223, 235, 246, 258, 271, 283, 296, 310, 323, 337, 352, 366, 381, 397, 412, ——
Table[Module[{t = k = 0}, While[t < n (n + 1)/2, k++; t += IntegerLength[k];]; k], {n, 50}]
{1},
{1, 2, 3},
{1, 2, 3, 4, 5, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, 4, 3, 4, 4, ...,4, 8, 4, 9, 5, 0},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, 4, 1, 4, 2, ...,2, 5, 3, 5, 4, 5, 5, 5, 6, 5, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, 3, 9, 4, 0, ...,7, 5, 8, 5, 9, 6, 0, 6, 1, 6, 2, 6, 3, 6, 4, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, 3, 7, 3, 8, ...,6, 3, 6, 4, 6, 5, 6, 6, 6, 7, 6, 8, 6, 9, 7, 0, 7, 1, 7, 2, 7},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, 3, 5, 3, 6, ...,6, 9, 7, 0, 7, 1, 7, 2, 7, 3, 7, 4, 7, 5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, 3, 3, 3, 4, ...,5, 7, 6, 7, 7, 7, 8, 7, 9, 8, 0, 8, 1, 8, 2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 0},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, 3, 1, 3, 2, ...,2, 8, 3, 8, 4, 8, 5, 8, 6, 8, 7, 8, 8, 8, 9, 9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, 2, 9, 3, 0, ...,9, 0, 9, 1, 9, 2, 9, 3, 9, 4, 9, 5, 9, 6, 9, 7, 9, 8, 9, 9, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 1, 0, 6},
{1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 0, 1, 1, 1, 2, 1, 3, 1, 4, 1, 5, 1, 6, 1, 7, 1, 8, 1, 9, 2, 0, 2, 1, 2, 2, 2, 3, 2, 4, 2, 5, 2, 6, 2, 7, 2, 8, ...,9, 8, 9, 9, 1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 0, 3, 1, 0, 4, 1, 0, 5, 1, 0, 6, 1, 0, 7, 1, 0, 8, 1, 0, 9, 1, 1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 3},
Table[First[RealDigits[ChampernowneNumber[], 10, n (n + 1)/2]], {n, 21}]
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
 楼主| 发表于 2025-8-24 11:29:25 | 显示全部楼层

题目1。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 24579, 235689, 2345789, 13456789, 123456789, 1234567899, 12344567899, 123345667899, 1233455677899, 12234456677899, 122334556678899, 1223344556778899, 12233445566778899, 112233445566778899}
Table[s = Sort[PadRight[{}, n^2, Range[9]]]; u = Table[{s[]}, {i, n}]; r = s[[n + 1 ;;]]; For[i = 1, i <= Length[r], i++, t = r[];  w = First[Ordering[FromDigits /@ u, 1]]; u[[w]] = Append[u[[w]], t];]; Max[FromDigits /@ u], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目2。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 7642, 77542, 886532, 8865432, 88765431, 987654321, 9876654321, 98776543321, 988765543321, 9887665443221, 98877655433221, 988876655433221, 9888776554433211, 98887766554433211, 998877665544332211}
Table[s = Reverse[Sort[PadRight[{}, n^2, Range[9]]]]; u = Table[{s[]}, {i, n}]; r = s[[n + 1 ;;]]; For[i = 1, i <= Length[r], i++, t = r[]; w = First[Ordering[FromDigits /@ u, 1]]; u[[w]] = Append[u[[w]], t];]; Min[FromDigits /@ u], {n, 1, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目3。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 23579, 234679, 2245789, 22345789, 123456789, 1023456789, 10234567899, 102344567899, 1023345667899, 10223445677899, 101233455677899, 1012334456678899, 10122344556778899, 101223344566778899}
Table[s = Sort[Mod[Range[n^2], 10]]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目4。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 6642, 77531, 875421, 8865421, 88754321, 887654321, 9876543210, 98765543210, 987765432210, 9877665433210, 98876554432210, 988766554332110, 9887766544322110, 98887665544322110, 988877655443322110}
Table[t = Sort[Mod[Range[n^2], 10], Greater][[1 ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, Sort[Mod[Range[n^2], 10], Greater][[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

这4个编码是不是搞复杂了?谢谢!!!
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
 楼主| 发表于 2025-8-24 14:54:17 | 显示全部楼层
题目1。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 24579, 235689, 2345789, 13456789, 123456789, 1234567899, 12344567899, 123345667899, 1233455677899, 12234456677899, 122334556678899, 1223344556778899, 12233445566778899, 112233445566778899}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目2。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 7642, 77542, 886532, 8865432, 88765431, 987654321, 9876654321, 98776543321, 988765543321, 9887665443221, 98877655433221, 988876655433221, 9888776554433211, 98887766554433211, 998877665544332211}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1, Greater]; t = s[[1 ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目3。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 23579, 234679, 2245789, 22345789, 123456789, 1023456789, 10234567899, 102344567899, 1023345667899, 10223445677899, 101233455677899, 1012334456678899, 10122344556778899, 101223344566778899}
Table[s = Sort[Mod[Range[n^2], 10]]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目4。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 6642, 77531, 875421, 8865421, 88754321, 887654321, 9876543210, 98765543210, 987765432210, 9877665433210, 98876554432210, 988766554332110, 9887766544322110, 98887665544322110, 988877655443322110}
Table[s = Sort[Mod[Range[n^2], 10], Greater]; t = s[[1 ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

这4个编码是不是搞复杂了?——这些按钮我还是学不好。谢谢!!!
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
 楼主| 发表于 2025-8-24 16:25:38 | 显示全部楼层
题目1。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 24579, 235689, 2345789, 13456789, 123456789, 1234567899, 12344567899, 123345667899, 1233455677899, 12234456677899, 122334556678899, 1223344556778899, 12233445566778899, 112233445566778899}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1]; v = s[[ ;; n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目2。A = {1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,6,7,8,9,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 7642, 77542, 886532, 8865432, 88765431, 987654321, 9876654321, 98776543321, 988765543321, 9887665443221, 98877655433221, 988876655433221, 9888776554433211, 98887766554433211, 998877665544332211}
Table[s = Sort[Mod[Range[0, n^2 - 1], 9] + 1, Greater]; t = s[[ ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目3。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最小时,其中最大的n位数=R(n)。R(n)是这样一串数。
{1, 24, 369, 2469, 23579, 234679, 2245789, 22345789, 123456789, 1023456789, 10234567899, 102344567899, 1023345667899, 10223445677899, 101233455677899, 1012334456678899, 10122344556778899, 101223344566778899}
Table[s = Sort[Mod[Range[n^2], 10]]; t = Count[s, 0]; v = s[[t + 1 ;; t + n]]; Do[w = Ordering[v, 1][[1]]; v[[w]] = v[[w]]*10 + d, {d, Join[s[[1 ;; t]], s[[t + n + 1 ;;]]]}]; Max[v], {n, 18}]
编码依据: 1,将A前n^2个数码进行升序排列。2,分配n个最小数码(非0)为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。

题目4。A = {1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,......},
用A前n^2个数码, 恰好可以组成n个n位数,当n个数乘积最大时,其中最小的n位数=R(n)。R(n)是这样一串数。
{1, 32, 763, 6642, 77531, 875421, 8865421, 88754321, 887654321, 9876543210, 98765543210, 987765432210, 9877665433210, 98876554432210, 988766554332110, 9887766544322110, 98887665544322110, 988877655443322110}
Table[s = Sort[Mod[Range[n^2], 10], Greater]; t = s[[ ;; n]]; Do[w = Ordering[t, 1][[1]]; t[[w]] = t[[w]]*10 + d, {d, s[[n + 1 ;;]]}]; Min[t], {n, 18}]
编码依据: 1,将A前n^2个数码进行降序排列。2,分配n个最大数码为最高位,使n个数每个都有一个当前值。3,每次进行相同的分配:依次取数码填在当前值最小的数后面。4,最后得到最小的n位数=R(n)。
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
 楼主| 发表于 2025-8-27 06:52:57 | 显示全部楼层
OEIS有这串数——{1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600, 6227020800, 87178291200, 1307674368000, 20922789888000}——没有这 3 个公式

Table[Permanent@Table[If[k - n < i < k + n, 1, 0], {k, n}, {i, n}], {n, 16}] == Table[Permanent@Table[If[k < i < k, 1, 1], {k, n}, {i, n}], {n, 16}] == Table[Permanent@Table[If[k < i < k, 0, 1], {k, n}, {i, n}], {n, 16}]
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
 楼主| 发表于 2025-8-27 07:13:18 | 显示全部楼层
A000166——{0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106}

Table[Permanent@Table[If[k ≤ i ≤ k, 0, 1], {k, n}, {i, n}], {n, 19}]————这通项公式不也挺好???
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
 楼主| 发表于 2025-8-27 08:46:09 | 显示全部楼层
好玩!!!——OEIS有这串数——A000271——{0, 0, 1, 3, 16, 96, 675, 5413, 48800, 488592, 5379333, 64595975, 840192288, 11767626752, 176574062535, 2825965531593}

可没有我们这么“好”的通项公式——Table[Permanent@Table[If[k ≤ i ≤ k + 1, 0, 1], {k, n}, {i, n}], {n, 16}]
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
 楼主| 发表于 2025-8-27 09:40:38 | 显示全部楼层
题目: 将 $1, 2, 3, ..., i, ..., n$ 这 n 个数重新排列,  得到新序列 $a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}$ 。 约定 $ i-A≤a_{i}≤i+B。A,B$ 是不同或相同的正整数。求满足条件的排列数量。

可以有统一的公式!Table[Permanent@Table[If[$k - A <= i <= k + B, 1, 0$], {k, n}, {i, n}], {n, 24}]——OEIS没有这样简单的公式。

下面的公式可以提速。$A,B$不变。——可惜公式变长了。

g[n_, A_, B_] := Module[{W = A + B + 1, p, u, s = 0, t, k, v, i, d},  For[k = 0, k < W, k++, If[(d = 1 - A + k) < 1 || d > n, s = BitOr[s, 2^k]]]; p = UnitVector[2^W, s + 1];

For[i = 1, i <= n, i++, u = ConstantArray[0, 2^W]; For[t = 0, t < 2^W, t++, If[p[[t + 1]] > 0, For[k = 0, k < W, k++, If[BitAnd[t, 2^k] == 0 && 1 <= (d = i - A + k) <= n,

v = BitShiftRight[BitOr[t, 2^k], 1]; If[i + B > n, v = BitOr[v, 2^(W - 1)]]; u[[v + 1]] += p[[t + 1]]]]]]; p = u];Total[p]]  f = Table[g[n, 3, 5], {n, 24}]

譬如——A=1,B=8 。A104144——有公式。
Table[Permanent@Table[If[k - 1 <= i <= k + 8, 1, 0], {k, n}, {i, n}], {n, 24}]
{1, 2, 4, 8, 16, 32, 64, 128, 256, 511, 1021, 2040, 4076, 8144, 16272, 32512, 64960, 129792, 259328, 518145, 1035269, 2068498, 4132920, 8257696, 16499120, 32965728, 65866496, 131603200, 262947072, 525375999, 1049716729}

譬如——A=2,B=6 。A072853——有个长长的公式。
Table[Permanent@Table[If[k - 2 <= i <= k + 6, 1, 0], {k, n}, {i, n}], {n, 21}]
{1, 2, 6, 18, 54, 162, 486, 1394, 3991, 11593, 33772, 98320, 286072, 831952, 2418664, 7030816, 20441944, 59441521, 172843609, 502580846, 1461344622, 4249102850, 12354982862, 35924300898, 104456501102, 303726483778},

譬如————A=3,B=5 。A072855——有个怪怪的公式。
Table[Permanent@Table[If[k - 3 <= i <= k + 5, 1, 0], {k, n}, {i, n}], {n, 16}]
{1, 2, 6, 24, 96, 384, 1374, 4718, 16275, 57749, 206756, 739780, 2637348, 9378840, 33318804, 118439044}

譬如————A=5,B=7 。A179347——没有公式了。
Table[Permanent@Table[If[k - 5 <= i <= k + 7, 1, 0], {k, n}, {i, n}], {n, 15}]
{1, 2, 6, 24, 120, 720, 4320, 25920, 140520, 714264, 3519294, 17234438, 85314915, 431525429, 2206564916}

譬如————A=5,B=11 。OEIS——没有这串数了。
Table[Permanent@Table[If[k - 5 <= i <= k + 11, 1, 0], {k, n}, {i, n}], {n, 15}]
{1, 2, 6, 24, 120, 720, 4320, 25920, 155520, 933120, 5598720, 33592320, 192178920, 1070271384, 5893613214}
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
 楼主| 发表于 2025-8-28 09:47:30 | 显示全部楼层
题目: 将 $1, 2, 3, ..., i, ..., n$ 这 n 个数重新排列,  得到新序列 $a_{1}, a_{2}, a_{3}, ..., a_{i}, ..., a _{n}$ 。 约定 $ i-1≤a_{i}≤i+B$ 。求满足条件的排列数量a(n)。

给出找这串具体数的一种方法——既不重复又不遗漏的去找。——又不搞复杂。

B=1,——n只能出现在尾部的2项。
{1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025,121393,196418,317811,514229, 832040,1346269}
先得有前2项——a(1)=1,{1},  a(2)=2,{1,2},{2,1},
a(3)=2(每个a(2)后面加3)+1((每个a(1)后面加3,2))=3,——a(3)=3,{1,2,3},{2,1,3},{1,3,2},——后面加3,2——顺序不能动!
a(4)=3(每个a(3)后面加4)+2((每个a(2)后面加4,3))=5,——a(4)=5,{1,2,3,4},{2,1,3,4},{1,3,2,4},{1,2,4,3},{2,1,4,3},
a(5)=5(每个a(4)后面加5)+3((每个a(3)后面加5,4))=8,——a(5)=8,{1,2,3,4,5},{2,1,3,4,5},{1,3,2,4,5},{1,2,4,3,5},{2,1,4,3,5},{1,2,3,5,4},{2,1,3,5,4},{1,3,2,5,4},
a(6)=8(每个a(5)后面加6)+5((每个a(4)后面加6,5))=13,——{1,2,3,4,5,6}{2,1,3,4,5,6}{1,3,2,4,5,6}{1,2,4,3,5,6}{2,1,4,3,5,6}{1,2,3,5,4,6}{2,1,3,5,4,6}{1,3,2,5,4,6},{1,2,3,4,6,5}{2,1,3,4,6,5}{1,3,2,4,6,5}{1,2,4,3,6,5}{2,1,4,3,6,5},
......
B=2,——n只能出现在尾部的3项。
{1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705, 3136, 5768, 10609, 19513, 35890, 66012, 121415, 223317, 410744, 755476, 1389537, 2555757, 4700770, 8646064}
先得有前3项——a(1)=1,{1},  a(2)=2,{1,2},{2,1},  a(3)=4,{1,2,3},{1,3,2},{2,1,3},{3,1,2},
a(4)=4(每个a(3)后面加4)+2((每个a(2)后面加4,3))+1((每个a(1)后面加4,2,3))=7,——a(4)=7,{1,2,3,4},{1,3,2,4},{2,1,3,4},{3,1,2,4},{1,2,4,3},{2,1,4,3},{1,4,2,3},——后面加4,2,3——顺序不能动!
a(5)=7(每个a(4)后面加5)+4((每个a(3)后面加5,4))+2((每个a(2)后面加5,3,4))=13,——{1,2,3,4,5}{1,3,2,4,5}{2,1,3,4,5}{3,1,2,4,5}{1,2,4,3,5}{2,1,4,3,5}{1,4,2,3,5},{1,2,3,5,4}{1,3,2,5,4}{2,1,3,5,4}{3,1,2,5,4},{1,2,5,3,4}{2,1,5,3,4},
a(6)=13(每个a(5)后面加6)+7((每个a(4)后面加6,5))+4((每个a(3)后面加6,4,5))=24,——a(6)=24,——注意尾项只有2种可能:6或5。
{1,2,3,4,5,6}{1,3,2,4,5,6}{2,1,3,4,5,6}{3,1,2,4,5,6}{1,2,4,3,5,6}{2,1,4,3,5,6}{1,4,2,3,5,6},{1,2,3,5,4,6}{1,3,2,5,4,6}{2,1,3,5,4,6}{3,1,2,5,4,6},{1,2,5,3,4,6}{2,1,5,3,4,6},
{1,2,3,4,6,5}{1,3,2,4,6,5}{2,1,3,4,6,5}{3,1,2,4,6,5}{1,2,4,3,6,5}{2,1,4,3,6,5}{1,4,2,3,6,5},{1,2,3,6,4,5}{1,3,2,6,4,5}{2,1,3,6,4,5}{3,1,2,6,4,5},
a(7)=24(每个a(6)后面加7)+13((每个a(5)后面加7,6))+7((每个a(4)后面加7,5,6))=44,——a(7)=44,——注意尾项只有2种可能:7或6。——后面加7,5,6——顺序不能动!
{1,2,3,4,5,6,7}{1,3,2,4,5,6,7}{2,1,3,4,5,6,7}{3,1,2,4,5,6,7}{1,2,4,3,5,6,7}{2,1,4,3,5,6,7}{1,4,2,3,5,6,7}{1,2,3,5,4,6,7}{1,3,2,5,4,6,7}{2,1,3,5,4,6,7}{3,1,2,5,4,6,7}{1,2,5,3,4,6,7}{2,1,5,3,4,6,7}
{1,2,3,4,6,5,7}{1,3,2,4,6,5,7}{2,1,3,4,6,5,7}{3,1,2,4,6,5,7}{1,2,4,3,6,5,7}{2,1,4,3,6,5,7}{1,4,2,3,6,5,7}{1,2,3,6,4,5,7}{1,3,2,6,4,5,7}{2,1,3,6,4,5,7}{3,1,2,6,4,5,7},
{1,2,3,4,5,7,6}{1,3,2,4,5,7,6}{2,1,3,4,5,7,6}{3,1,2,4,5,7,6}{1,2,4,3,5,7,6}{2,1,4,3,5,7,6}{1,4,2,3,5,7,6},{1,2,3,5,4,7,6}{1,3,2,5,4,7,6}{2,1,3,5,4,7,6}{3,1,2,5,4,7,6}{1,2,5,3,4,7,6}{2,1,5,3,4,7,6},
{1,2,3,4,7,5,6}{1,3,2,4,7,5,6}{2,1,3,4,7,5,6}{3,1,2,4,7,5,6}{1,2,4,3,7,5,6}{2,1,4,3,7,5,6}{1,4,2,3,7,5,6},
......
B=3,——n只能出现在尾部的4项。
{1, 2, 4, 8, 15, 29, 56, 108, 208, 401, 773, 1490, 2872, 5536, 10671, 20569, 39648, 76424, 147312, 283953, 547337, 1055026, 2033628, 3919944, 7555935, 14564533}
先得有前4项——a(1)=1,{1},  a(2)=2,{1,2},{2,1},  a(3)=4,{1,2,3},{1,3,2},{2,1,3},{3,1,2},  a(4)=8,{1,2,3,4},{1,2,4,3},{1,3,2,4},{1,4,2,3},{2,1,3,4},{2,1,4,3},{3,1,2,4},{4,1,2,3},
a(5)=8(每个a(4)后面加5)+4((每个a(3)后面加5,4))+2((每个a(2)后面加5,3,4))+1((每个a(1)后面加5,2,3,4))=15——注意尾项只有2种可能:5或4。——后面加5,2,3,4——顺序不能动!
{1,2,3,4,5}{1,2,4,3,5}{1,3,2,4,5}{1,4,2,3,5}{2,1,3,4,5}{2,1,4,3,5}{3,1,2,4,5}{4,1,2,3,5},{1,2,3,5,4}{1,3,2,5,4}{2,1,3,5,4}{3,1,2,5,4},{1,2,5,3,4}{2,1,5,3,4},{1,5,2,3,4},
a(6)=15(每个a(5)后面加6)+8((每个a(4)后面加6,5))+4((每个a(3)后面加6,4,5))+2((每个a(2)后面加6,3,4,5))=29,——注意尾项只有2种可能:6或5。——后面加6,3,4,5——顺序不能动!
{1,2,3,4,5,6}{1,2,4,3,5,6}{1,3,2,4,5,6}{1,4,2,3,5,6}{2,1,3,4,5,6}{2,1,4,3,5,6}{3,1,2,4,5,6}{4,1,2,3,5,6}{1,2,3,5,4,6}{1,3,2,5,4,6}{2,1,3,5,4,6}{3,1,2,5,4,6}{1,2,5,3,4,6}{2,1,5,3,4,6}{1,5,2,3,4,6},
{1,2,3,4,6,5}{1,2,4,3,6,5}{1,3,2,4,6,5}{1,4,2,3,6,5}{2,1,3,4,6,5}{2,1,4,3,6,5}{3,1,2,4,6,5}{4,1,2,3,6,5},{1,2,3,6,4,5}{1,3,2,6,4,5}{2,1,3,6,4,5}{3,1,2,6,4,5},{1,2,6,3,4,5}{2,1,6,3,4,5},
a(7)=29(每个a(6)后面加7)+15((每个a(5)后面加7,6))+8((每个a(4)后面加7,5,6))+4((每个a(3)后面加7,4,5,6))=56,——注意尾项只有2种可能:7或6。——中间的数也是蛮有规律的。
a(8)=56(每个a(7)后面加8)+29((每个a(6)后面加8,7))+15((每个a(5)后面加8,6,7))+8((每个a(4)后面加8,5,6,7))=108,——注意尾项只有2种可能:8或7。
......
B=4,——n只能出现在尾部的5项。
{1, 2, 4, 8, 16, 31, 61, 120, 236, 464, 912, 1793, 3525, 6930, 13624, 26784, 52656, 103519, 203513, 400096, 786568, 1546352, 3040048, 5976577, 11749641, 23099186}
先得有前5项——a(1)=1,{1},  a(2)=2,{1,2},{2,1},  a(3)=4,{1,2,3},{1,3,2},{2,1,3},{3,1,2},  a(4)=8,{1,2,3,4},{1,2,4,3},{1,3,2,4},{1,4,2,3},{2,1,3,4},{2,1,4,3},{3,1,2,4},{4,1,2,3},  a(5)=16,——... 。
a(6)=16(每个a(5)后面加6)+8((每个a(4)后面加6,5))+4((每个a(3)后面加6,4,5))+2((每个a(2)后面加6,3,4,5))+1((每个a(1)后面加6,2,3,4,5))=31,——注意尾项只有2种可能:6或5。——后面加6,2,3,4,5——顺序不能动!
{1,2,3,4,5,6}{1,2,3,5,4,6}{1,2,4,3,5,6}{1,2,5,3,4,6}{1,3,2,4,5,6}{1,3,2,5,4,6}{1,4,2,3,5,6},{1,5,2,3,4,6},{2,1,3,4,5,6}{2,1,3,5,4,6}{2,1,4,3,5,6}{2,1,5,3,4,6},{3,1,2,4,5,6}{3,1,2,5,4,6}{4,1,2,3,5,6}{5,1,2,3,4,6},
{1,2,3,4,6,5}{1,2,4,3,6,5}{1,3,2,4,6,5}{1,4,2,3,6,5}{2,1,3,4,6,5}{2,1,4,3,6,5}{3,1,2,4,6,5}{4,1,2,3,6,5},{1,2,3,6,4,5}{1,3,2,6,4,5}{2,1,3,6,4,5}{3,1,2,6,4,5},{1,2,6,3,4,5}{2,1,6,3,4,5},{1,6,2,3,4,5},
a(7)=31(每个a(6)后面加7)+16((每个a(5)后面加7,6))+8((每个a(4)后面加7,5,6))+4((每个a(3)后面加7,4,5,6))+2((每个a(2)后面加7,3,4,5,6))=61,——注意尾项只有2种可能:7或6。
a(8)=61(每个a(7)后面加8)+31((每个a(6)后面加8,7))+16((每个a(5)后面加8,6,7))+8((每个a(4)后面加8,5,6,7))+4((每个a(3)后面加8,4,5,6,7))=120,——注意尾项只有2种可能:8或7。
a(9)=120(每个a(8)后面加9)+61((每个a(7)后面加9,8))+31((每个a(6)后面加9,7,8))+16((每个a(5)后面加9,6,7,8))+8((每个a(4)后面加9,5,6,7,8))=236,——注意尾项只有2种可能:9或8。
......
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
 楼主| 发表于 2025-8-28 14:37:58 | 显示全部楼层
A000217——0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 105, 120, 136, 153, 171, 190, 210, 231, 253, ——应该有下面的图。看着这个图——可以有很多想法。
                                   
                                 66,
                              67,65,
                            68,36,64,
                        69,37,35,63,
                     70,38,15,34,62,99,
                  71,39,16,14,33,61,98,  
               72,40,17,03,13,32,60,97,
            73,41,18,04,02,12,31,59,96,
         74,42,19,05,00,01,11,30,58,95,
      75,43,20,06,07,08,09,10,29,57,94,
    76,44,21,22,23,24,25,26,27,28,56,93,
  77,45,46,47,48,49,50,51,52,53,54,55,92,
78,79,80,81,82,83,84,85,86,87,88,89,90,91,

OEIS有这个图吗?——从中央00-01-03-06-10-15-21-28-36-45-55-66-78-91,...这些是拐弯点。如何画这个图?
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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