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

1,1

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(2,1) 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).

Links

Table of n, a(n) for n=1..57.

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.

Example

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,...).

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 *)

Crossrefs

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

Sequence in context: A173657 A213358 A065935 * A166669 A286701 A380590

Adjacent sequences: A361972 A361973 A361974 * A361976 A361977 A361978

Keyword

nonn,tabl

Author

Clark Kimberling, Apr 01 2023

Status

approved