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%I #28 Mar 20 2013 12:31:09
%S 1,1,0,1,1,0,1,2,0,0,1,3,2,0,0,1,4,5,0,0,0,0,5,9,5,0,0,0,0,5,14,14,0,
%T 0,0,0,0,0,19,28,14,0,0,0,0,0,0,19,47,42,0,0,0,0,0,0,0,0,66,89,42,0,0,
%U 0,0,0,0,0,0,66,155,131,0,0,0,0,0,0,0,0,0,0,221,286,131,0,0
%N Square array T, read by antidiagonals: T(n,k) = 0 if n-k >= 1 or if k-n >= 6, T(0,0) = T(0,1) = T(0,2) = T(0,3) = T(0,4) = T(0,5) = 1, T(n,k) = T(n-1,k) + T(n,k-1).
%C A hexagon arithmetic of E. Lucas.
%D E. Lucas, Théorie des nombres, A.Blanchard, Paris, 1958, Tome 1, p.89
%F T(n,n) = A080937(n).
%F T(n,n+1) = A080937(n+1).
%F T(n,n+2) = A094790(n+1).
%F T(n,n+3) = A094789(n+1).
%F T(n,n4) = T(n,n+5) = A005021(n).
%F Sum_{k, 0<=k<=n} T(n-k,k) = A028495(n).
%e Square array begins:
%e 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=0
%e 0, 1, 2, 3, 4, 5, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, ... row n=1
%e 0, 0, 2, 5, 9, 14, 19, 19, 0, 0, 0, 0, 0, 0, 0, ... row n=2
%e 0, 0, 0, 5, 14, 28, 47, 66, 66, 0, 0, 0, 0, 0, 0, ... row n=3
%e 0, 0, 0, 0, 14, 42, 89, 155, 221, 221, 0, 0, 0, 0, ... row n=4
%e 0, 0, 0, 0, 0, 0, 42, 131, 286, 507, 728, 728, 0, 0, ... row n=5
%e 0, 0, 0, 0, 0, 0, 131, 417, 924, 1652, 2380, 2380, 0, ... row n=6
%e ...
%t Clear[t]; t[0, k_ /; k <= 5] = 1; t[n_, k_] /; k < n || k > n+5 = 0; t[n_, k_] := t[n, k] = t[n-1, k] + t[n, k-1]; Table[t[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* _Jean-François Alcover_, Mar 18 2013 *)
%Y Cf. A006053, A052547, A096976, A187066,
%Y Cf. Similar sequences A216230, A216228, A216226, A216238
%K nonn,tabl
%O 0,8
%A _Philippe Deléham_, Mar 16 2013
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