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%I #31 Jul 22 2019 03:24:52
%S 0,0,0,0,-1,-1,0,1,0,0,0,0,1,0,0,0,2,2,3,2,2,0,1,3,3,4,3,3,0,3,4,6,6,
%T 7,6,6,0,2,5,6,8,8,9,8,8,0,4,6,9,10,12,12,13,12,12,0,3,7,9,12,13,15,
%U 15,16,15,15,0,5,8,12,14,17,18,20,20,21,20,20
%N A(n, k) is a square array read in the decreasing antidiagonals, for n >= 0 and k >= 0.
%F Let h(n) = 0, 0, -1, A198442(1), A198442(2), A198442(3), ... Then A(n, 0) = h(n), A(n, 1) = h(n+1) and A(n, k) = A(n, k-2) + n otherwise.
%e The array starts:
%e [n\k][0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, ...]
%e [0] 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... = A000004
%e [1] 0, -1, 1, 0, 2, 1, 3, 2, 4, 3, 5, 4, ... = A028242(n-2)
%e [2] -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... = A023443(n)
%e [3] 0, 0, 3, 3, 6, 6, 9, 9, 12, 12, 15, 15, ... = 3*A004526(n)
%e [4] 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, ... = A005843(n)
%e [5] 2, 3, 7, 8, 12, 13, 17, 18, 22, 23, 27, 28, ... = A047221(n+1)
%e [6] 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, ... = A008585(n+1)
%e [7] 6, 8, 13, 15, 20, 22, 27, 29, 34, 36, 41, 43, ... = A047336(n+2)
%e [8] 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, ... = A008586(n+2)
%e Successive columns: A198442(n-2), A198442(n-1), A004652(n), A198442(n+1), A198442(n+2), A079524(n), ... .
%e First subdiagonal: 0, 0, 3, 6, ... = A242477(n).
%e First upperdiagonal: 0, 1, 2, 6, 10, ... = A238377(n-1).
%e Array written as a triangle:
%e 0;
%e 0, 0;
%e 0, -1, -1;
%e 0, 1, 0, 0;
%e 0, 0, 1, 0, 0;
%e 0, 2, 2, 3, 2, 2;
%e etc.
%p A := proc(n, k) option remember; local h;
%p h := n -> `if`(n<3, [0, 0, -1][n+1], iquo(n^2-4*n+3, 4));
%p if k = 0 then h(n) elif k = 1 then h(n+1) else A(n, k-2) + n fi end: # _Peter Luschny_, Sep 08 2018
%t h[n_] := If[n < 3, {0, 0, -1}[[n + 1]], Quotient[n^2 - 4 n + 3, 4]];
%t A[n_, k_] := A[n, k] = If[k == 0, h[n], If[k == 1, h[n+1], A[n, k-2] + n]];
%t Table[A[n - k, k], {n, 0, 11}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Jul 22 2019, after _Peter Luschny_ *)
%Y Cf. A000004, A004526, A004652, A005843, A008585, A008586, A023443, A028242, A047221, A047336, A079524, A198442, A238377, A242477.
%K sign,tabl
%O 0,17
%A _Paul Curtz_, Sep 06 2018
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