A361976 (2,2)-block array, B(2,2), of the natural number array (A000027), read by descending antidiagonals.

11, 31, 39, 67, 75, 83, 119, 127, 135, 143, 187, 195, 203, 211, 219, 271, 279, 287, 295, 303, 311, 371, 379, 387, 395, 403, 411, 419, 487, 495, 503, 511, 519, 527, 535, 543, 619, 627, 635, 643, 651, 659, 667, 675, 683, 767, 775, 783, 791, 799, 807, 815, 823

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(2,2) is a row-splitting array. The rows and columns of B(2,2) are linearly recurrent with signature (3,-3,1). It appears that the order array (as defined in A333029) of B(2,2) is given by A000027.

Examples

			Corner of B(2,2):
   11   31   67   119   187   271
   39   75  127   195   279   379
   83  135  203   287   387   503
  143  211  295   395   511   643
  219  303  403   519   651   799
		

Crossrefs

Cf. A000027, A333029, A361974 (array B(1,2)), A361975 (array B(2,1)).

Programs

Mathematica

zz = 10; z = 13;
w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
t[n_, k_] := w[2 n - 1, 2 k - 1] + w[2 n - 1, 2 k] + w[2 n, 2 k - 1] + w[2 n, 2 k]
Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361976 sequence*)
TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361976 array*)

Formula

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

b(i,j) = 8(i+j)^2 - 12i - 20 j + 11.