A376499 Array read by ascending antidiagonals: A(n,k) = A376484/(2*n+1).

1, 1, 3, 1, 3, 9, 1, 3, 10, 27, 1, 3, 10, 35, 81, 1, 3, 10, 35, 125, 243, 1, 3, 10, 35, 126, 450, 729, 1, 3, 10, 35, 126, 462, 1625, 2187, 1, 3, 10, 35, 126, 462, 1715, 5875, 6561, 1, 3, 10, 35, 126, 462, 1716, 6419, 21250, 19683, 1, 3, 10, 35, 126, 462, 1716, 6435, 24157, 76875, 59049

Offset

1,3

Comments

It is only a conjecture that the A(n,k) are always integers.

Values repeated as a staircase for all A(n+x,2*n) (x > 0 and are equal to A(n,2*n)).

Links

Table of n, a(n) for n=1..66.

Formula

A(n,k) = A376484(n,k)/(2*n+1)

Example

First ten rows start as follows:
  1 3  9 27  81 243  729 2187  6561 19683  59049  177147  531441  1594323  4782969
  1 3 10 35 125 450 1625 5875 21250 76875 278125 1006250 3640625 13171875 47656250
  1 3 10 35 126 462 1715 6419 24157 91238 345401 1309574 4970070 18874261 71705865
  1 3 10 35 126 462 1716 6435 24309 92358 352485 1350054 5185350 19960020 76964985
  1 3 10 35 126 462 1716 6435 24310 92378 352715 1352054 5199975 20055024 77531355
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200299 20058272 77558325
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558759
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760
  1 3 10 35 126 462 1716 6435 24310 92378 352716 1352078 5200300 20058300 77558760

Prog

(C++) double gen(int n, int m) {
    double s = 0, d = 1;
    for(int i = 1; i <= m; i++) d *= 4;
    for(int i = 1; i <= n; i++) {
        double v = 1;
        for(int j = 1; j <= 2 * m; j++) v *= sin(2 * i * M_PI / (2 * n + 1));
        s += v * d;
    }
    return s / (2 * n + 1);
}

Crossrefs

All of these are conjectures. Rows: A000244, A081567, A122068. Columns: A000012, A000012 * 3, A095049 for n >= 20. A(1,k) = A000244, A(2,k) = A081567, A(3,k) = A122068 (First 3 rows of the array).A(n,1) = A(n,2) / 3 = A000012, A(n,3) = A095049 for n >= 20 (First 3 columns of the array). When k increases, the row of A(n,k) gets closer to A001700.

Sequence in context: A010282 A119265 A143453 * A248830 A350562 A164308

Adjacent sequences: A376496 A376497 A376498 * A376500 A376501 A376502

Keyword

nonn,tabl

Author

Cheng-Jun Li, Sep 25 2024

Status

approved