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A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).

%I A368343 #25 Dec 22 2023 16:06:47
%S A368343 1,1,1,1,2,1,1,3,3,2,1,4,7,5,2,1,5,13,16,7,2,1,6,21,41,34,9,3,1,7,31,
%T A368343 86,125,70,12,3,1,8,43,157,346,377,143,15,3,1,9,57,260,787,1386,1134,
%U A368343 289,18,4,1,10,73,401,1562,3937,5547,3405,581,22,4
%N A368343 Square array T(n,k), n >= 3, k >= 0, read by antidiagonals downwards, where T(n,k) = Sum_{j=0..n} k^(n-j) * floor(j/3).
%H A368343 Seiichi Manyama, <a href="/A368343/b368343.txt">Antidiagonals n = 3..142, flattened</a>
%F A368343 T(n,k) = T(n-3,k) + Sum_{j=0..n-3} k^j.
%F A368343 T(n,k) = 1/(k-1) * Sum_{j=0..n} floor(k^j/(k^2+k+1)) = Sum_{j=0..n} floor(k^j/(k^3-1)) for k > 1.
%F A368343 T(n,k) = (k+1)*T(n-1,k) - k*T(n-2,k) + T(n-3,k) - (k+1)*T(n-4,k) + k*T(n-5,k).
%F A368343 G.f. of column k: x^3/((1-x) * (1-k*x) * (1-x^3)).
%F A368343 T(n,k) = 1/(k-1) * (floor(k^(n+1)/(k^3-1)) - floor((n+1)/3)) for k > 1.
%e A368343 Square array begins:
%e A368343   1,  1,   1,    1,    1,     1,     1, ...
%e A368343   1,  2,   3,    4,    5,     6,     7, ...
%e A368343   1,  3,   7,   13,   21,    31,    43, ...
%e A368343   2,  5,  16,   41,   86,   157,   260, ...
%e A368343   2,  7,  34,  125,  346,   787,  1562, ...
%e A368343   2,  9,  70,  377, 1386,  3937,  9374, ...
%e A368343   3, 12, 143, 1134, 5547, 19688, 56247, ...
%o A368343 (PARI) T(n, k) = sum(j=0, n, k^(n-j)*(j\3));
%Y A368343 Columns k=0..4 give A002264, A130518, A178455, A368344, A368345.
%Y A368343 Cf. A055129, A368296.
%K A368343 nonn,tabl
%O A368343 3,5
%A A368343 _Seiichi Manyama_, Dec 22 2023