A289992 Number of magic labelings of the prism graph I X C_8 having magic sum n.
1, 49, 746, 6122, 34067, 144963, 506772, 1524628, 4074949, 9898229, 22220990, 46695870, 92769495, 175610631, 318756136, 557659432, 944355593, 1553488697, 2489980818, 3898657938, 5976186139, 8985711691, 13274641084, 19296041660, 27634190285
Offset: 0
Links
- R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973. [Cached copy, with permission]
- Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
Formula
From Colin Barker, Sep 13 2017: (Start)
G.f.: (1 + x)*(1 + 6*x + x^2)*(1 + 32*x + 70*x^2 + 32*x^3 + x^4) / (1 - x)^10.
a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n>9. (End)
[Proof of the g.f. follows from the convolution formula and insertion of the g.f. A244880(x): Sum_{n>=0} a(n)x^n = Sum_{n>=0} A244880(n)*x^n +2*Sum_{n>=0} Sum_{i=0..n-1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} Sum_{n>=i+1} A244880(i)*x^n = A244880(x) +2*Sum_{i>=0} A244880(i)*x^(i+1) Sum_{n>=0} x^n = A244880(x)+2*A244880(x)*x/(1+x) = A244880(x)*(1+2*x/(1-x)). R. J. Mathar, Mar 09 2025]