一道趣题

nyy

王守恩 发表于 2025-5-15 10:47
谢谢 wayne！应该是这个。谢谢 wayne！

Table[R = Round[Sqrt[2] T]; {(R - R^2 + 2 T^2)/2, (R + R^2 - ...

我认为你应该用Floor函数，而不是Round函数

iseemu2009

通项公式的求法

nyy

王守恩 发表于 2025-5-14 12:16
{1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172, 191, 211, 232, 254, 27 ...

我来根据你的通项公式，然后由T得到(m,n)的通项公式

1. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
2. (*假设数T在第一行,先算出(1,t),t向下取整*)
   
3. ans=Solve[{
   
4.     ((m+n)^2-m-3n+2)/2
   
5.     ==T/.{m->1}
   
6. },{n}]
   
7. t=Floor[n]/.ans[[2]]
   
8. aa=T-((m+n)^2-m-3n+2)/2/.{m->1,n->t} (*算出T与(1,t)所对应的差值*)
   
9. {m,n}={1+aa,t-aa}//FullSimplify (*根据差值移动(1,t)到(m,n),这样就是通项公式*)
   
10. bb=Table[{m,n,T},{T,50}](*检验一下通项公式*)
    

复制代码

求解结果
\[\left\{\frac{1}{2} \left(-\left\lfloor \frac{1}{2} \left(\sqrt{8 T-7}+1\right)\right\rfloor ^2+\left\lfloor \frac{1}{2} \left(\sqrt{8 T-7}+1\right)\right\rfloor +2 T\right),\frac{1}{2} \left(\left\lfloor \frac{1}{2} \left(\sqrt{8 T-7}+1\right)\right\rfloor ^2+\left\lfloor \frac{1}{2} \left(\sqrt{8 T-7}+1\right)\right\rfloor -2 T+2\right)\right\}\]

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

nyy

nyy 发表于 2025-5-16 10:37
我来根据你的通项公式，然后由T得到(m,n)的通项公式

1. Clear["Global`*"];(*mathematica11.2,win7(64bit)Clear all variables*)
   
2. (*假设数T在第一行,先算出(1,t),t向下取整*)
   
3. ans=Solve[{
   
4.     ((m+n)^2-m-3n+2)/2
   
5.     ==T/.{m->1}
   
6. },{n}]
   
7. t=Floor[n]/.ans[[2]]
   
8. aa=T-((m+n)^2-m-3n+2)/2/.{m->1,n->t} (*算出T与(1,t)所对应的差值*)
   
9. {m,n}={1+aa,t-aa}//FullSimplify (*根据差值移动(1,t)到(m,n),这样就是通项公式*)
   
10. bb=Table[{m,n,T},{T,50}](*检验一下通项公式*)
    
11. f=Function[{T},{#1,#2}]/.{#1->m,#2->n}(*定义函数*)
    
12. MatrixForm[SparseArray[Table[f[k]->Style[k,Blue,20],{k,50}]]]
    

复制代码

得到结果

nyy

nyy 发表于 2025-5-16 10:37
我来根据你的通项公式，然后由T得到(m,n)的通项公式

谁能把上面的取下整放在表达式的最外面？
我觉得放在表达式的里面太丑了！

iseemu2009

求T在第几行第几列详细过程

iseemu2009

上面的详细过程最终用编程就是如下代码，T值可任意修改，求T在数列的第几行第几列：

1. T = 26;       k = Ceiling[(Sqrt[8 T + 1] - 1)/2];
   
2. {m = 1 + T - (k^2 - k + 2)/2, n = k - T + (k^2 - k + 2)/2}

复制代码

王守恩

iseemu2009 发表于 2025-5-16 12:24
上面的详细过程最终用编程就是如下代码，T值可任意修改，求T在数列的第几行第几列：
...

这个与你是等价的。

Table[k = Round[Sqrt[2 T]]; {k - k^2 + 2 T, k + k^2 - 2 T + 2}/2, {T, 26}]

{{1, 1}, {1, 2}, {2, 1}, {1, 3}, {2, 2}, {3, 1}, {1, 4}, {2, 3}, {3, 2}, {4, 1}, {1, 5}, {2, 4}, {3, 3}, {4, 2}, {5, 1}, {1, 6}, {2, 5}, {3, 4}, {4, 3}, {5, 2}, {6, 1}, {1, 7}, {2, 6}, {3, 5}, {4, 4}, {5, 3}}

iseemu2009

求新数列的通项公式

王守恩

iseemu2009 发表于 2025-5-16 15:37
求新数列的通项公式

{1, 2, 6, 7, 15, 16, 28, 29, 45, 46, 66, 67, 91, 92, 120, 121, 153, 154, 190, 191, 231, 232, 276, 277, 325, 326, 378, 379, 435, 436, 496, 497, 561, 562, 630, 631, 703, 704, 780, 781},
{3, 5, 8, 14, 17, 27, 30, 44, 47, 65, 68, 90, 93, 119, 122, 152, 155, 189, 192, 230, 233, 275, 278, 324, 327, 377, 380, 434, 437, 495, 498, 560, 563, 629, 632, 702, 705, 779, 782, 860},
{4, 9, 13, 18, 26, 31, 43, 48, 64, 69, 89, 94, 118, 123, 151, 156, 188, 193, 229, 234, 274, 279, 323, 328, 376, 381, 433, 438, 494, 499, 559, 564, 628, 633, 701, 706, 778, 783, 859, 864},
{10, 12, 19, 25, 32, 42, 49, 63, 70, 88, 95, 117, 124, 150, 157, 187, 194, 228, 235, 273, 280, 322, 329, 375, 382, 432, 439, 493, 500, 558, 565, 627, 634, 700, 707, 777, 784, 858, 865, 943},
{11, 20, 24, 33, 41, 50, 62, 71, 87, 96, 116, 125, 149, 158, 186, 195, 227, 236, 272, 281, 321, 330, 374, 383, 431, 440, 492, 501, 557, 566, 626, 635, 699, 708, 776, 785, 857, 866, 942, 951},
{21, 23, 34, 40, 51, 61, 72, 86, 97, 115, 126, 148, 159, 185, 196, 226, 237, 271, 282, 320, 331, 373, 384, 430, 441, 491, 502, 556, 567, 625, 636, 698, 709, 775, 786, 856, 867, 941, 952, 1030},
{22, 35, 39, 52, 60, 73, 85, 98, 114, 127, 147, 160, 184, 197, 225, 238, 270, 283, 319, 332, 372, 385, 429, 442, 490, 503, 555, 568, 624, 637, 697, 710, 774, 787, 855, 868, 940, 953, 1029, 1042},
{36, 38, 53, 59, 74, 84, 99, 113, 128, 146, 161, 183, 198, 224, 239, 269, 284, 318, 333, 371, 386, 428, 443, 489, 504, 554,  569, 623, 638, 696, 711, 773, 788, 854, 869, 939, 954, 1028, 1043, 1121},
{37, 54, 58, 75, 83, 100, 112, 129, 145, 162, 182, 199, 223, 240, 268, 285, 317, 334, 370, 387, 427, 444, 488, 505, 553, 570, 622, 639, 695, 712, 772, 789, 853, 870, 938, 955, 1027, 1044, 1120, 1137}

Table[((m + n) (m + n - 2) - Cos[(m - n) Pi] (m - n) + 2)/2, {m, 9}, {n, 40}]