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;]; 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, 10, n (n + 1)/2]], {n, 21}]
northwolves 发表于 2025-7-27 08:14
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]]; u = Table[{s[]}, {i, n}]; r = s[]; For, i++, t = r[];w = First@ u, 1]]; u[] = Append], t];]; Max, {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]]]; u = Table[{s[]}, {i, n}]; r = s[]; For, i++, t = r[]; w = First]; u[] = Append], t];]; Min, {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, 10]]; t = Count; v = s[]; Do[]; v[] = v[]*10 + d, {d, Join], s[]]}]; Max, {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, 10], Greater][]; Do[]; t[] = t[]*10 + d, {d, Sort, 10], Greater][]}]; Min, {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！！！
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, 9] + 1]; t = Count; v = s[]; Do[]; v[] = v[]*10 + d, {d, Join], s[]]}]; Max, {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, 9] + 1, Greater]; t = s[]; Do[]; t[] = t[]*10 + d, {d, s[]}]; Min, {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, 10]]; t = Count; v = s[]; Do[]; v[] = v[]*10 + d, {d, Join], s[]]}]; Max, {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, 10], Greater]; t = s[]; Do[]; t[] = t[]*10 + d, {d, s[]}]; Min, {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！！！
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, 9] + 1]; v = s[[ ;; n]]; Do[]; v[] = v[]*10 + d, {d, s[]}]; Max, {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, 9] + 1, Greater]; t = s[[ ;; n]]; Do[]; t[] = t[]*10 + d, {d, s[]}]; Min, {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, 10]]; t = Count; v = s[]; Do[]; v[] = v[]*10 + d, {d, Join], s[]]}]; Max, {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, 10], Greater]; t = s[[ ;; n]]; Do[]; t[] = t[]*10 + d, {d, s[]}]; Min, {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)。
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, {k, n}, {i, n}], {n, 16}] == Table, {k, n}, {i, n}], {n, 16}] == Table, {k, n}, {i, n}], {n, 16}]
A000166——{0, 1, 2, 9, 44, 265, 1854, 14833, 133496, 1334961, 14684570, 176214841, 2290792932, 32071101049, 481066515734, 7697064251745, 130850092279664, 2355301661033953, 44750731559645106}
Table, {k, n}, {i, n}], {n, 19}]————这通项公式不也挺好？？？
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, {k, n}, {i, n}], {n, 16}]
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, {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 := Module[{W = A + B + 1, p, u, s = 0, t, k, v, i, d},For]]; p = UnitVector;
For; For] > 0, For == 0 && 1 <= (d = i - A + k) <= n,
v = BitShiftRight, 1]; If]; u[] += p[]]]]]; p = u];Total]f = Table, {n, 24}]
譬如——A=1,B=8 。A104144——有公式。
Table, {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, {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, {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, {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, {k, n}, {i, n}], {n, 15}]
{1, 2, 6, 24, 120, 720, 4320, 25920, 155520, 933120, 5598720, 33592320, 192178920, 1070271384, 5893613214}
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。
......
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？