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%I #7 Jun 19 2022 02:20:48
%S 1,0,3,0,2,9,0,1,8,27,0,0,6,26,81,0,0,4,23,80,243,0,0,3,20,76,242,729,
%T 0,0,3,17,72,237,728,2187,0,0,1,17,66,232,722,2186,6561,0,0,1,11,66,
%U 222,716,2179,6560,19683,0,0,1,11,54,222,701,2172,6552,19682,59049
%N Square array A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1, read by antidiagonals.
%H G. C. Greubel, <a href="/A061980/b061980.txt">Antidiagonals n = 0..50, flattened</a>
%F A(n,k) = A(n-1,k) + A(n-1, floor(k/2)) + A(n-1, floor(k/3)), with A(0,0) = 1.
%F T(n, k) = A(k, n-k).
%F Sum_{k=0..n} A(n, k) = A000400(n).
%F T(n, n) = A(n, 0) = A000244(n). - _G. C. Greubel_, Jun 18 2022
%e Array begins as:
%e 1, 0, 0, 0, 0, 0, 0, ...;
%e 3, 2, 1, 0, 0, 0, 0, ...;
%e 9, 8, 6, 4, 3, 3, 1, ...;
%e 27, 26, 23, 20, 17, 17, 11, ...;
%e 81, 80, 76, 72, 66, 66, 54, ...;
%e 243, 242, 237, 232, 222, 222, 202, ...;
%e 729, 728, 722, 716, 701, 701, 671, ...;
%e Antidiagonal rows begin as:
%e 1;
%e 0, 3;
%e 0, 2, 9;
%e 0, 1, 8, 27;
%e 0, 0, 6, 26, 81;
%e 0, 0, 4, 23, 80, 243;
%e 0, 0, 3, 20, 76, 242, 729;
%e 0, 0, 3, 17, 72, 237, 728, 2187;
%e 0, 0, 1, 17, 66, 232, 722, 2186, 6561;
%t A[n_, k_]:= A[n, k]= If[n==0, Boole[k==0], A[n-1,k] +A[n-1,Floor[k/2]] +A[n-1, Floor[k/3]]];
%t T[n_, k_]:= A[k, n-k];
%t Table[A[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Jun 18 2022 *)
%o (SageMath)
%o @CachedFunction
%o def A(n,k):
%o if (n==0): return 0^k
%o else: return A(n-1, k) + A(n-1, (k//2)) + A(n-1, (k//3))
%o def T(n, k): return A(k, n-k)
%o flatten([[T(n, k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 18 2022
%Y Row sums are 6^n: A000400.
%Y Columns are A000244, A024023, A060188, A061981, A061982 twice, A061983 twice, etc.
%Y Cf. A000244, A061290, A061930, A061979, A061984, A061987.
%K nonn,tabl
%O 0,3
%A _Henry Bottomley_, May 24 2001
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