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A361974 (1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.

%I A361974 #8 Jun 04 2023 23:50:42
%S A361974 3,11,8,27,20,15,51,40,31,24,83,68,55,44,35,123,104,87,72,59,48,171,
%T A361974 148,127,108,91,76,63,227,200,175,152,131,112,95,80,291,260,231,204,
%U A361974 179,156,135,116,99,363,328,295,264,235,208,183,160,139,120,443,404
%N A361974 (1,2)-block array, B(1,2), of the natural number array (A000027), read by descending antidiagonals.
%C A361974 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(1,2) is a row-splitting array. The rows and columns of B(1,2) are linearly recurrent with signature (3,-3,1).  It appears that the order array (as defined in A333029) of B(1,2) is given by A163255.
%F A361974 B(1,2) = (b(i,j)), where b(i,j) = w(i, 2j-1) + w(i, 2j) for i >= 1, j >=         1, where (w(i,j)) is the natural number array (A000027).
%F A361974 b(i,j) = 2i + (i + 2j - 2)^2.
%e A361974 Corner of B(1,2):
%e A361974    3   11   27   51   83  123   171   227
%e A361974    8   20   40   68  104  148   200   260
%e A361974   15   31   55   87  127  175   231   295
%e A361974   24   44   72  108  152  204   264   332
%e A361974   35   59   91  131  179  235   299   371
%e A361974   48   76  112  156  298  268   336   412
%e A361974 (row 1 of A000027) = (1,2,4,7,11,16,22,29,...), so (row 1 of B(1,2)) = (3,11,27,58,...);
%e A361974 (row 2 of A000027) = (3,5,8,12,17,23,30,38,...), so (row 2 of B(1,2)) = (8,20,40,68,...).
%t A361974 zz = 10; z = 13;
%t A361974 w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
%t A361974 t[h_, k_] := w[h, 2 k - 1] + w[h, 2 k];
%t A361974 Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361974 sequence*)
%t A361974 TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361974 array*)
%Y A361974 Cf. A000027, A163255, A333029, A361975 (array B(2,1)), A361976 (array B(2,2)).
%K A361974 nonn,tabl
%O A361974 1,1
%A A361974 _Clark Kimberling_, Apr 01 2023