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

1,1

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.

Links

Table of n, a(n) for n=1..53.

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.

Example

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

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*)

Crossrefs

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

Sequence in context: A022423 A173972 A167488 * A298566 A090756 A038351

Adjacent sequences: A361973 A361974 A361975 * A361977 A361978 A361979

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 01 2023

Status

approved