A203286 Number of arrays of 2n nondecreasing integers in -3..3 with sum zero and equal numbers greater than zero and less than zero.

4, 12, 28, 57, 104, 176, 280, 425, 620, 876, 1204, 1617, 2128, 2752, 3504, 4401, 5460, 6700, 8140, 9801, 11704, 13872, 16328, 19097, 22204, 25676, 29540, 33825, 38560, 43776, 49504, 55777, 62628, 70092, 78204, 87001, 96520, 106800, 117880, 129801

Offset: 1

Author

R. H. Hardin, Dec 31 2011

Keywords

• nonn

Comments

Column 3 of A203291.

a(n-4) seems to be the number of face-magic cubes or order 2 with magic sum n, which means the sum of the 4 numbers at the 4 corners of each of the 6 faces equals n. (The 8 integers at the corners do not need to be distinct; copies by the 48 operations of rotations and flips are counted separately. All 8 integers are positive.). E.g., 4 =a(5-4) is the number of cubes with magic sum 5 obtained by placing 1 at 6 of the 8 corners but 2 at two corners opposite to each other along a space diagonal (with 4 different space diagonals available). See also A381589 and A115264. - R. J. Mathar, Mar 11 2025

Examples

			Some solutions for n=3:
.-2...-2...-2...-2...-3...-3...-3...-3...-1...-3....0...-2...-1...-3...-2...-3
..0...-2...-2...-1....0...-3...-1...-1...-1...-2....0...-2...-1...-1...-2...-2
..0...-2....0...-1....0...-2....0...-1...-1...-1....0....0....0...-1...-1...-2
..0....1....0....1....0....2....0....1....1....1....0....0....0....1....1....2
..0....2....1....1....0....3....2....2....1....2....0....2....1....1....2....2
..2....3....3....2....3....3....2....2....1....3....0....2....1....3....2....3
		

Links

• R. H. Hardin, Table of n, a(n) for n = 1..210

Crossrefs

Cf. A203291, A005994.

Formula

Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6).

Conjectures from Colin Barker, Jun 04 2018: (Start)

G.f.: x*(4 - 4*x + 5*x^3 - 4*x^4 + x^5) / ((1 - x)^5*(1 + x)).

a(n) = (48 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n even.

a(n) = (42 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n odd.

(End)