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Number of magic labelings of the prism graph I X C_5 with magic sum n.
13

%I #18 Aug 04 2021 15:18:10

%S 1,11,57,197,533,1223,2494,4654,8105,13355,21031,31891,46837,66927,

%T 93388,127628,171249,226059,294085,377585,479061,601271,747242,920282,

%U 1123993,1362283,1639379,1959839,2328565,2750815,3232216,3778776,4396897,5093387,5875473,6750813,7727509,8814119

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

%C 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

%H Colin Barker, <a href="/A244497/b244497.txt">Table of n, a(n) for n = 0..1000</a>

%H R. P. Stanley, <a href="/A002721/a002721.pdf">Examples of Magic Labelings</a>, Unpublished Notes, 1973 [Cached copy, with permission]

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,5,5,-9,5,-1).

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

%F From _Colin Barker_, Jan 13 2017: (Start)

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

%F 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.

%F (End)

%p 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

%t 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 *)

%t LinearRecurrence[{5,-9,5,5,-9,5,-1},{1,11,57,197,533,1223,2494},40] (* _Harvey P. Dale_, Aug 04 2021 *)

%o (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

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

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

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_, Jul 07 2014