cp's OEIS Frontend

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

%I A361975 #16 Jun 07 2023 04:40:20
%S A361975 4,7,16,12,23,36,19,32,47,64,28,43,60,79,100,39,56,75,96,119,144,52,
%T A361975 71,92,115,140,167,196,67,88,111,136,163,192,223,256,84,107,132,159,
%U A361975 188,219,252,287,324,103,128,155,184,215,248,283,320,359,400,124,151
%N A361975 (2,1)-block array, B(2,1), of the natural number array (A000027), read by descending antidiagonals.
%C A361975 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).
%F A361975 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).
%F A361975 b(i,j) = 4i - 1 + (2i + j - 2)^2.
%e A361975 Corner of B(2,1):
%e A361975     4    7   12   19   28   39   52
%e A361975    16   23   32   43   56   71   88
%e A361975    36   47   60   75   92  111  132
%e A361975    64   79   96  115  136  159  184
%e A361975   100  119  140  163  188  215  244
%e A361975   144  167  192  219  238  279  312
%e A361975 (column 1 of A000027) = (1,3,6,10,15,21,...), so (column 1 of B(2,1)) = (4,16,64,...);
%e A361975 (column 2 of A000027) = (2,5,9,14,20,27,...), so (column 2 of B(2,1)) = (7,23,47,...).
%t A361975 zz = 10; z = 13;
%t A361975 w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
%t A361975 t[h_, k_] := w[2 h - 1, k] + w[2 h, k];
%t A361975 Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (* this sequence *)
%t A361975 TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (* this sequence as an array *)
%Y A361975 Cf. A000027, A056537, A333029, A361974 (array B(1,2)), A361976 (array B(2,2)).
%K A361975 nonn,tabl
%O A361975 1,1
%A A361975 _Clark Kimberling_, Apr 01 2023