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%I #19 Mar 07 2023 05:55:21
%S 16,105,496,1759,5052,12469,27412,55059,102952,181543,304908,491563,
%T 765184,1155567,1699684,2442553,3438468,4752283,6460432,8652429,
%U 11432392,14920189,19253232,24588229,31102456,38995845,48492976,59844451,73329300
%N Number of triangular of a 5 X 5 X 5 0..n arrays with all rows and diagonals having the same length having the same sum, with corners zero.
%H R. H. Hardin, <a href="/A195806/b195806.txt">Table of n, a(n) for n = 1..32</a>
%H M. Kauers and C. Koutschan, <a href="https://arxiv.org/abs/2303.02793">Some D-finite and some possibly D-finite sequences in the OEIS</a>, arXiv:2303.02793 [cs.SC], 2023.
%F From _Manuel Kauers_ and _Christoph Koutschan_, Mar 01 2023: (Start)
%F Conjectured recurrence: a(n) - 3*a(n+1) + 2*a(n+2) - a(n+3) + 6*a(n+4) - 5*a(n+5) - 3*a(n+6) + 3*a(n+8) + 5*a(n+9) - 6*a(n+10) + a(n+11) - 2*a(n+12) + 3*a(n+13) - a(n+14) = 0.
%F Conjectured closed form as a quasi-polynomial:
%F a(6*n) = 1 + 25*n + 158*n^2 + 650*n^3 + 2275*n^4 + 4680*n^5 + 4680*n^6.
%F a(6*n+1) = 16 + 198*n + 1133*n^2 + 3900*n^3 + 8125*n^4 + 9360*n^5 + 4680*n^6.
%F a(6*n+2) = 105 + 1087*n + 4922*n^2 + 12350*n^3 + 17875*n^4 + 14040*n^5 + 4680*n^6.
%F a(6*n+3) = 496 + 4148*n + 14783*n^2 + 28600*n^3 + 31525*n^4 + 18720*n^5 + 4680*n^6.
%F a(6*n+4) = 1759 + 12121*n + 35258*n^2 + 55250*n^3 + 49075*n^4 + 23400*n^5 + 4680*n^6.
%F a(6*n+5) = (1+n)^2*(5052 + 19370*n + 28405*n^2 + 18720*n^3 + 4680*n^4). (End)
%e Some solutions for n=4:
%e 0 0 0 0 0 0 0
%e 0 1 2 2 1 1 1 4 4 2 4 1 0 0
%e 2 0 2 1 0 4 0 3 0 4 2 0 2 4 2 1 0 4 3 2 3
%e 1 0 0 0 3 3 0 0 1 3 3 1 2 0 4 3 2 4 4 4 2 3 0 2 0 2 2 0
%e 0 0 2 1 0 0 1 1 4 0 0 1 0 1 0 0 3 2 2 0 0 4 2 2 0 0 3 1 3 0 0 0 3 0 0
%Y Row 5 of A195805.
%K nonn
%O 1,1
%A _R. H. Hardin_, Sep 23 2011
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