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%I #14 Jun 13 2015 00:51:17
%S 1,5,91,1125,17259,253125,3806091,56953125,854518059,12814453125,
%T 192222105291,2883251953125,43248906698859,648731689453125,
%U 9730978399444491,145964630126953125,2189469525287839659,32842041778564453125,492630628439671823691,7389459400177001953125
%N Central term in powers of the Lo-Shu Magic Square as a matrix.
%C a(n)/a(n-1) tends to 15, the "Magic Number" of the Lo-Shu Magic Square.
%C There are a total of 8 variations of the Lo-Shu magic square by rotations and/or reflections. Four of the variations (those with 4, 5, 6 or 6, 5, 4 in the diagonal), have a(2) = 91. The other 4 variations (those with 2, 5, 8 or 8, 5, 2 in the diagonal - lower left to upper right - have a(2) = 59, but otherwise, a(n) for the latter sequence (central term in analogous powers of those matrices) = A091281(n).
%C a(2k+1) = (5)*[15^(2k)]. E.g. a(5) = 253125 = (5)*(15^4).
%H <a href="/index/Rec">Index entries for linear recurrences with constant coefficients</a>, signature (15,24,-360).
%F The Lo-Shu magic square square as a 3 X 3 matrix is: [8, 1, 6, / 3, 5, 7 / 4, 9, 2] = M. Then a(n) = central term in M^n.
%F (1/69) {23*15^n - 2*24^[(n+1)/2] + 2*24^[(n+2)/2] }. - _Ralf Stephan_, Dec 02 2004
%F G.f.: -(8*x^2+10*x-1) / ((15*x-1)*(24*x^2-1)). [_Colin Barker_, Dec 10 2012]
%e a(2) = 91 since M^2 = [ 91, 67, 67 / 67, 91, 67, / 67, 67, 91]
%o (PARI) a(n)=([8,1,6;3,5,7;4,9,2]^n)[2,2] \\ _Charles R Greathouse IV_, Dec 14 2011
%Y Cf. A033812.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Dec 28 2003
%E a(12)-a(19) from _Charles R Greathouse IV_, Dec 14 2011
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