A125198 Number of magical labelings of the octahedral graph of magic sum n.

1, 8, 40, 144, 417, 1032, 2272, 4568, 8545, 15072, 25320, 40824, 63553, 95984, 141184, 202896, 285633, 394776, 536680, 718784, 949729, 1239480, 1599456, 2042664, 2583841, 3239600, 4028584, 4971624, 6091905, 7415136, 8969728, 10786976, 12901249, 15350184

Offset: 0

Author

R. J. Mathar, Jan 25 2007

Keywords

• easy
• nonn

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• M. M. Ahmed, Algebraic Combinatorics of Magic Squares, math.CO/0405476, p73.
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
• Index entries for linear recurrences with constant coefficients, signature (6,-14,14,0,-14,14,-6,1).

Programs

Maple

a := proc(r) local r2 ; r2 := r^6/120+r^5/10+25*r^4/48+3*r^3/2+38*r^2/15+12*r/5 ; if r mod 2 = 0 then r2+1 ; else r2+15/16 ; fi ; end: for n from 0 to 40 do printf("%d ",a(n)) ; od;

Mathematica

(1 + 2*x + 6*x^2 + 2*x^3 + x^4)/((1 - x)^7*(1 + x)) + O[x]^40 // CoefficientList[#, x]& (* Jean-François Alcover, Apr 01 2018 *)

PARI

Vec((1+2*x+6*x^2+2*x^3+x^4)/((1-x)^7*(1+x)) + O(x^40)) \\ Colin Barker, Jan 13 2017

Formula

G.f.: (1+2*x+6*x^2+2*x^3+x^4)/((1-x)^7*(1+x)). [Stanley] - N. J. A. Sloane, Jul 07 2014

From Colin Barker, Jan 13 2017: (Start)

a(n) = (15*(31+(-1)^n) + 1152*n + 1216*n^2 + 720*n^3 + 250*n^4 + 48*n^5 + 4*n^6) / 480.

a(n) = 6*a(n-1) - 14*a(n-2) + 14*a(n-3) - 14*a(n-5) + 14*a(n-6) - 6*a(n-7) + a(n-8) for n>7.

(End)

Extensions

Stanley reference added by N. J. A. Sloane, Jul 07 2014