A244497 Number of magic labelings of the prism graph I X C_5 with magic sum n.

1, 11, 57, 197, 533, 1223, 2494, 4654, 8105, 13355, 21031, 31891, 46837, 66927, 93388, 127628, 171249, 226059, 294085, 377585, 479061, 601271, 747242, 920282, 1123993, 1362283, 1639379, 1959839, 2328565, 2750815, 3232216, 3778776, 4396897, 5093387, 5875473, 6750813, 7727509, 8814119

Offset: 0

Author

N. J. A. Sloane, Jul 07 2014

Keywords

• nonn
• easy

Comments

This sequence is also the number of magic labelings of the cycle-of-loops graph LOOP X C_5 with magic sum n, where LOOP is the 1-vertex, 1-loop-edge graph. A similar identity holds between the sequences for I X C_k and LOOP X C_k for all odd k. - David J. Seal, Sep 14 2017

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• R. P. Stanley, Examples of Magic Labelings, Unpublished Notes, 1973 [Cached copy, with permission]
• Index entries for linear recurrences with constant coefficients, signature (5,-9,5,5,-9,5,-1).

Crossrefs

Cf. A019298, A061927, A292281, A244873, A289992 (analogs for prism graphs I X C_k, k = 3,4,6,7,8).

Cf. A006325, A244879, A244880 (analogs for LOOP X C_k, k = 4,6,8).

Programs

Maple

A244497:=n->(3*(63+(-1)^n) + 576*n + 720*n^2 + 460*n^3 + 150*n^4 + 20*n^5) / 192: seq(A244497(n), n=0..50); # Wesley Ivan Hurt, Sep 16 2017

Mathematica

CoefficientList[Series[(1 + 6 x + 11 x^2 + 6 x^3 + x^4)/((1 - x)^6*(1 + x)), {x, 0, 37}], x] (* Michael De Vlieger, Sep 15 2017 *)
LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,11,57,197,533,1223,2494},40] (* Harvey P. Dale, Aug 04 2021 *)

PARI

Vec((1+6*x+11*x^2+6*x^3+x^4) / ((1-x)^6*(1+x)) + O(x^40)) \\ Colin Barker, Jan 13 2017

Formula

G.f.: (1 + 6*x + 11*x^2 + 6*x^3 + x^4) / ((1 - x)^6*(1 + x)).

From Colin Barker, Jan 13 2017: (Start)

a(n) = (3*(63+(-1)^n) + 576*n + 720*n^2 + 460*n^3 + 150*n^4 + 20*n^5) / 192.

a(n) = 5*a(n-1) - 9*a(n-2) + 5*a(n-3) + 5*a(n-4) - 9*a(n-5) + 5*a(n-6) - a(n-7) for n>6.

(End)