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Number of zero-sum -n..n arrays of 4 elements with adjacent element differences also in -n..n.
2

%I #11 Mar 03 2018 05:35:49

%S 7,31,81,171,309,509,779,1133,1579,2131,2797,3591,4521,5601,6839,8249,

%T 9839,11623,13609,15811,18237,20901,23811,26981,30419,34139,38149,

%U 42463,47089,52041,57327,62961,68951,75311,82049,89179,96709,104653,113019

%N Number of zero-sum -n..n arrays of 4 elements with adjacent element differences also in -n..n.

%C Row 4 of A202252.

%H R. H. Hardin, <a href="/A202254/b202254.txt">Table of n, a(n) for n = 1..210</a>

%F Empirical: a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).

%F Conjectures from _Colin Barker_, Mar 03 2018: (Start)

%F G.f.: x*(7 + 10*x + 2*x^2 + 4*x^3 - x^4) / ((1 - x)^4*(1 + x)).

%F a(n) = (22*n^3 + 33*n^2 + 26*n + 12) / 12 for n even.

%F a(n) = (22*n^3 + 33*n^2 + 26*n + 3) / 12 for n odd.

%F (End)

%e Some solutions for n=10:

%e 1 -1 1 -7 -7 -5 5 0 2 -6 4 -2 -2 9 -3 -9

%e 5 -8 7 1 1 -3 1 -5 -1 -2 6 3 1 1 1 1

%e -1 1 1 6 5 1 -4 1 -2 8 0 4 -4 0 6 0

%e -5 8 -9 0 1 7 -2 4 1 0 -10 -5 5 -10 -4 8

%Y Cf. A202252.

%K nonn

%O 1,1

%A _R. H. Hardin_, Dec 14 2011