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

1, 3, 8, 16, 31, 51, 80, 118, 167, 227, 302, 390, 495, 617, 758, 918, 1101, 1305, 1534, 1788, 2069, 2377, 2716, 3084, 3485, 3919, 4388, 4892, 5435, 6015, 6636, 7298, 8003, 8751, 9546, 10386, 11275, 12213, 13202, 14242, 15337, 16485, 17690, 18952, 20273, 21653

Offset: 0

Author

R. H. Hardin Mar 21 2011

Keywords

• nonn

Comments

Row 4 of A188122.

Examples

			Some solutions for n=6
.-6...-7...-8...-8...-5...-7...-6...-6...-7...-5...-8...-4...-5...-7...-7...-4
.-1...-2...-5...-2...-4...-2...-4...-4...-6...-4....1...-3...-2...-6...-3...-3
..3....4....5....2....2....1....4....3....6....4....2....3...-1....5....3....2
..4....5....8....8....7....8....6....7....7....5....5....4....8....8....7....5
a(0) = 1 with unique solution [-2, -1, 1, 2]. - _Michael Somos_, Apr 11 2011
		

Links

• R. H. Hardin, Table of n, a(n) for n = 0..200 (corrected by _R. H. Hardin_, Jan 19 2019)

Programs

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 */

Formula

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.

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

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