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A199899 Number of -n..n arrays x(0..3) of 4 elements with zero sum, and adjacent elements not both strictly positive and not both strictly negative.

%I A199899 #8 Feb 23 2018 07:28:24
%S A199899 15,49,111,209,351,545,799,1121,1519,2001,2575,3249,4031,4929,5951,
%T A199899 7105,8399,9841,11439,13201,15135,17249,19551,22049,24751,27665,30799,
%U A199899 34161,37759,41601,45695,50049,54671,59569,64751,70225,75999,82081,88479,95201
%N A199899 Number of -n..n arrays x(0..3) of 4 elements with zero sum, and adjacent elements not both strictly positive and not both strictly negative.
%C A199899 Row 4 of A199898.
%H A199899 R. H. Hardin, <a href="/A199899/b199899.txt">Table of n, a(n) for n = 1..200</a>
%F A199899 Empirical: a(n) = (4/3)*n^3 + 6*n^2 + (20/3)*n + 1.
%F A199899 Conjectures from _Colin Barker_, Feb 23 2018: (Start)
%F A199899 G.f.: x*(3 - x)*(5 - 2*x + x^2) / (1 - x)^4.
%F A199899 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n>4.
%F A199899 (End)
%e A199899 Some solutions for n=6:
%e A199899 ..3....3....4...-2....5...-2....5...-3....4...-3....0....2....0....6....3....1
%e A199899 ..0...-6...-4....6...-4....1...-5....2...-5....6....2...-2....5...-1...-5...-5
%e A199899 ..2....3....1...-6....3...-4....3...-1....5....0....0....5...-5....0....6....0
%e A199899 .-5....0...-1....2...-4....5...-3....2...-4...-3...-2...-5....0...-5...-4....4
%Y A199899 Cf. A199898.
%K A199899 nonn
%O A199899 1,1
%A A199899 _R. H. Hardin_, Nov 11 2011