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

0,2

Comments

Row 4 of A188122.

Links

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

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

Example

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

Prog

(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 */

Crossrefs

Sequence in context: A227265 A295960 A068039 * A081661 A005103 A001978

Adjacent sequences: A188120 A188121 A188122 * A188124 A188125 A188126

Keyword

nonn

Author

R. H. Hardin Mar 21 2011

Status

approved