A125196 Number of magic labelings of the Petersen graph with magic sum n.

1, 6, 27, 87, 228, 513, 1034, 1914, 3315, 5440, 8541, 12921, 18942, 27027, 37668, 51428, 68949, 90954, 118255, 151755, 192456, 241461, 299982, 369342, 450983, 546468, 657489, 785869, 933570, 1102695, 1295496

Offset: 0

Author

R. J. Mathar, Jan 25 2007

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
• M. M. Ahmed, Algebraic Combinatorics of Magic Squares, math.CO/0405476.
• Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Programs

Maple

a := proc(r) local r1 ; r1 := r^5/24+5*r^4/16+25*r^3/24+15*r^2/8+23*r/12 ; if r mod 2 = 0 then r1+1 ; else r1+13/16 ; fi ; end: for n from 0 to 30 do printf("%d ",a(n)) ; od;

Mathematica

CoefficientList[Series[(x^4 + x^3 + 6x^2 + x + 1)/(1 - x)^6/(1 + x), {x, 0, 40}], x] (* Vincenzo Librandi, Dec 12 2012 *)
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,6,27,87,228,513,1034},40] (* Harvey P. Dale, Sep 10 2024 *)

Formula

a(n) = (1/32)*(29*C(n+5,5) + 21*C(n+4,5) + 126*C(n+3,5) - 34*C(n+2,5) + 21*C(n+1,5) - 3*C(n,5) + 3*(-1)^n). [Stanley]. - N. J. A. Sloane, Jul 07 2014

G.f.: (x^4+x^3+6x^2+x+1)/((1-x)^6*(1+x)) [Stanley; Ahmed].

Extensions

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