OFFSET
1,1
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
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..210
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)
EXAMPLE
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
CROSSREFS
KEYWORD
nonn
AUTHOR
R. H. Hardin, Dec 31 2011
STATUS
approved