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A202785 Number of 3 X 3 0..n arrays with row and column sums equal.

%I A202785 #12 May 02 2019 16:03:39
%S A202785 14,87,340,1001,2442,5215,10088,18081,30502,48983,75516,112489,162722,
%T A202785 229503,316624,428417,569790,746263,964004,1229865,1551418,1936991,
%U A202785 2395704,2937505,3573206,4314519,5174092,6165545,7303506,8603647
%N A202785 Number of 3 X 3 0..n arrays with row and column sums equal.
%C A202785 Row 3 of A202784.
%H A202785 R. H. Hardin, <a href="/A202785/b202785.txt">Table of n, a(n) for n = 1..210</a>
%H A202785 Robert Israel, <a href="/A202785/a202785.pdf">Proof of empirical formula for A202785</a>
%H A202785 Wikipedia, <a href="https://en.wikipedia.org/wiki/Ehrhart_polynomial">Ehrhart polynomial</a>
%F A202785 Empirical: a(n) = (3/10)*n^5 + (3/2)*n^4 + (7/2)*n^3 + (9/2)*n^2 + (16/5)*n + 1.
%F A202785 Conjectures from _Colin Barker_, Jun 01 2018: (Start)
%F A202785 G.f.: x*(7 - 2*x + x^2)*(2 + x + 4*x^2 - x^3) / (1 - x)^6.
%F A202785 a(n) = 6*a(n-1) - 15*a(n-2) + 20*a(n-3) - 15*a(n-4) + 6*a(n-5) - a(n-6) for n>6.
%F A202785 (End)
%F A202785 Empirical formula verified (see link): _Robert Israel_, May 02 2019
%e A202785 Some solutions for n=7:
%e A202785 ..3..2..1....3..5..5....0..6..2....0..7..5....4..2..1....5..6..0....1..6..1
%e A202785 ..2..0..4....5..6..2....2..1..5....6..1..5....3..2..2....0..4..7....5..2..1
%e A202785 ..1..4..1....5..2..6....6..1..1....6..4..2....0..3..4....6..1..4....2..0..6
%p A202785 seq((3/10)*n^5 + (3/2)*n^4 + (7/2)*n^3 + (9/2)*n^2 + (16/5)*n + 1, n=1..30); # _Robert Israel_, May 02 2019
%Y A202785 Cf. A202784.
%K A202785 nonn
%O A202785 1,1
%A A202785 _R. H. Hardin_, Dec 24 2011