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A188123 Number of strictly increasing arrangements of 4 nonzero numbers in -(n+2)..(n+2) with sum zero.

%I A188123 #21 Jan 19 2019 07:23:12
%S A188123 1,3,8,16,31,51,80,118,167,227,302,390,495,617,758,918,1101,1305,1534,
%T A188123 1788,2069,2377,2716,3084,3485,3919,4388,4892,5435,6015,6636,7298,
%U A188123 8003,8751,9546,10386,11275,12213,13202,14242,15337,16485,17690,18952,20273,21653
%N A188123 Number of strictly increasing arrangements of 4 nonzero numbers in -(n+2)..(n+2) with sum zero.
%C A188123 Row 4 of A188122.
%H A188123 R. H. Hardin, <a href="/A188123/b188123.txt">Table of n, a(n) for n = 0..200</a> (corrected by _R. H. Hardin_, Jan 19 2019)
%F A188123 Empirical: a(n)=2*a(n-1)-a(n-3)-a(n-4)+2*a(n-6)-a(n-7) = 35/36 +2*n^2/3 +7*n/6 +2*n^3/9 +(-1)^n/4 -2*A049347(n)/9.
%F A188123 Empirical: G.f. -x*(-3-2*x-2*x^3-2*x^5+x^6) / ( (1+x)*(1+x+x^2)*(x-1)^4 ). - _R. J. Mathar_, Mar 21 2011
%F A188123 Empirical: a(n) = 1/108*(8*sqrt(3)*sin((2*Pi*n)/3) + 3*(2*n*(4*n*(n+3)+21) + 9*i*sin(Pi*n) + 35) - 24*cos((2*Pi*n)/3) + 27*cos(Pi*n)). - _Alexander R. Povolotsky_, Mar 21 2011
%e A188123 Some solutions for n=6
%e A188123 .-6...-7...-8...-8...-5...-7...-6...-6...-7...-5...-8...-4...-5...-7...-7...-4
%e A188123 .-1...-2...-5...-2...-4...-2...-4...-4...-6...-4....1...-3...-2...-6...-3...-3
%e A188123 ..3....4....5....2....2....1....4....3....6....4....2....3...-1....5....3....2
%e A188123 ..4....5....8....8....7....8....6....7....7....5....5....4....8....8....7....5
%e A188123 a(0) = 1 with unique solution [-2, -1, 1, 2]. - _Michael Somos_, Apr 11 2011
%o A188123 (PARI) {a(n) = local(v, c, m); m = n+2; forvec( v = vector( 4, i, [-m, m]), if( 0==prod( k=1, 4, v[k]), next); if( 0==sum( k=1, 4, v[k]), c++), 2); c} /* _Michael Somos_, Apr 11 2011 */
%K A188123 nonn
%O A188123 0,2
%A A188123 _R. H. Hardin_ Mar 21 2011