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

%I A203286 #17 Mar 13 2025 06:00:29
%S A203286 4,12,28,57,104,176,280,425,620,876,1204,1617,2128,2752,3504,4401,
%T A203286 5460,6700,8140,9801,11704,13872,16328,19097,22204,25676,29540,33825,
%U A203286 38560,43776,49504,55777,62628,70092,78204,87001,96520,106800,117880,129801
%N A203286 Number of arrays of 2n nondecreasing integers in -3..3 with sum zero and equal numbers greater than zero and less than zero.
%C A203286 Column 3 of A203291.
%C A203286 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
%H A203286 R. H. Hardin, <a href="/A203286/b203286.txt">Table of n, a(n) for n = 1..210</a>
%F A203286 Empirical: a(n) = 4*a(n-1) -5*a(n-2) +5*a(n-4) -4*a(n-5) +a(n-6).
%F A203286 Conjectures from _Colin Barker_, Jun 04 2018: (Start)
%F A203286 G.f.: x*(4 - 4*x + 5*x^3 - 4*x^4 + x^5) / ((1 - x)^5*(1 + x)).
%F A203286 a(n) = (48 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n even.
%F A203286 a(n) = (42 + 80*n + 52*n^2 + 16*n^3 + 2*n^4)/48 for n odd.
%F A203286 (End)
%e A203286 Some solutions for n=3:
%e A203286 .-2...-2...-2...-2...-3...-3...-3...-3...-1...-3....0...-2...-1...-3...-2...-3
%e A203286 ..0...-2...-2...-1....0...-3...-1...-1...-1...-2....0...-2...-1...-1...-2...-2
%e A203286 ..0...-2....0...-1....0...-2....0...-1...-1...-1....0....0....0...-1...-1...-2
%e A203286 ..0....1....0....1....0....2....0....1....1....1....0....0....0....1....1....2
%e A203286 ..0....2....1....1....0....3....2....2....1....2....0....2....1....1....2....2
%e A203286 ..2....3....3....2....3....3....2....2....1....3....0....2....1....3....2....3
%Y A203286 Cf. A203291, A005994.
%K A203286 nonn
%O A203286 1,1
%A A203286 _R. H. Hardin_, Dec 31 2011