A361975 (2,1)-block array, B(2,1), of the natural number array (A000027), read by descending antidiagonals.

4, 7, 16, 12, 23, 36, 19, 32, 47, 64, 28, 43, 60, 79, 100, 39, 56, 75, 96, 119, 144, 52, 71, 92, 115, 140, 167, 196, 67, 88, 111, 136, 163, 192, 223, 256, 84, 107, 132, 159, 188, 219, 252, 287, 324, 103, 128, 155, 184, 215, 248, 283, 320, 359, 400, 124, 151

Offset: 1

Author

Clark Kimberling, Apr 01 2023

Keywords

• nonn
• tabl

Comments

We begin with a definition. Suppose that W = (w(i,j)), where i >= 1 and j >= 1, is an array of numbers such that if m and n satisfy 1 <= m < n, then there exists k such that w(m,k+h) < w(n,h+1) < w(m,k+h+1) for every h >= 0. Then W is a row-splitting array. The array B(2,1) is a row-splitting array. The rows and columns of B(2,1) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(2,1) is given by A056537 (the dispersion of the nonsquares).

Examples

			Corner of B(2,1):
    4    7   12   19   28   39   52
   16   23   32   43   56   71   88
   36   47   60   75   92  111  132
   64   79   96  115  136  159  184
  100  119  140  163  188  215  244
  144  167  192  219  238  279  312
(column 1 of A000027) = (1,3,6,10,15,21,...), so (column 1 of B(2,1)) = (4,16,64,...);
(column 2 of A000027) = (2,5,9,14,20,27,...), so (column 2 of B(2,1)) = (7,23,47,...).
		

Crossrefs

Cf. A000027, A056537, A333029, A361974 (array B(1,2)), A361976 (array B(2,2)).

Programs

Mathematica

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[h_, k_] := w[2 h - 1, k] + w[2 h, k];
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* this sequence as an array *)

Formula

B(2,1) = (b(i,j)), where b(i,j) = w(2i-1, j) + w(2i, j) for i >= 1, j >= 1, where (w(i,j)) is the natural number array (A000027).

b(i,j) = 4i - 1 + (2i + j - 2)^2.