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

%I A361976 #4 Apr 01 2023 22:04:48
%S A361976 11,31,39,67,75,83,119,127,135,143,187,195,203,211,219,271,279,287,
%T A361976 295,303,311,371,379,387,395,403,411,419,487,495,503,511,519,527,535,
%U A361976 543,619,627,635,643,651,659,667,675,683,767,775,783,791,799,807,815,823
%N A361976 (2,2)-block array, B(2,2), of the natural number array (A000027), read by descending antidiagonals.
%C A361976 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.
%F A361976 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).
%F A361976 b(i,j) = 8(i+j)^2 - 12i - 20 j + 11.
%e A361976 Corner of B(2,2):
%e A361976    11   31   67   119   187   271
%e A361976    39   75  127   195   279   379
%e A361976    83  135  203   287   387   503
%e A361976   143  211  295   395   511   643
%e A361976   219  303  403   519   651   799
%t A361976 zz = 10; z = 13;
%t A361976 w[n_, k_] := n + (n + k - 2) (n + k - 1)/2;
%t A361976 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]
%t A361976 Table[t[n - k + 1, k], {n, 12}, {k, n, 1, -1}] // Flatten (*A361976 sequence*)
%t A361976 TableForm[Table[t[h, k], {h, 1, zz}, {k, 1, z}]] (*A361976 array*)
%Y A361976 Cf. A000027, A333029, A361974 (array B(1,2)), A361975 (array B(2,1)).
%K A361976 nonn,tabl
%O A361976 1,1
%A A361976 _Clark Kimberling_, Apr 01 2023