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A091281 Central term in powers of the Lo-Shu Magic Square as a matrix.

%I A091281 #24 Apr 06 2025 08:44:51
%S A091281 1,5,91,1125,17259,253125,3806091,56953125,854518059,12814453125,
%T A091281 192222105291,2883251953125,43248906698859,648731689453125,
%U A091281 9730978399444491,145964630126953125,2189469525287839659,32842041778564453125,492630628439671823691,7389459400177001953125
%N A091281 Central term in powers of the Lo-Shu Magic Square as a matrix.
%C A091281 a(n)/a(n-1) tends to 15, the "Magic Number" of the Lo-Shu Magic Square.
%C A091281 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).
%H A091281 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,24,-360).
%F A091281 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 A091281 a(2*k+1) = 5*15^(2*k). E.g. a(5) = 253125 = 5*15^4.
%F A091281 a(n) = (1/69)*(23*15^n - 2*24^floor((n+1)/2) + 2*24^floor((n+2)/2)). - _Ralf Stephan_, Dec 02 2004
%F A091281 G.f.: -(8*x^2+10*x-1) / ((15*x-1)*(24*x^2-1)). - _Colin Barker_, Dec 10 2012
%e A091281 a(2) = 91 since M^2 = [91, 67, 67 / 67, 91, 67 / 67, 67, 91].
%o A091281 (PARI) a(n)=([8,1,6;3,5,7;4,9,2]^n)[2,2] \\ _Charles R Greathouse IV_, Dec 14 2011
%Y A091281 Cf. A033812.
%K A091281 nonn,easy
%O A091281 0,2
%A A091281 _Gary W. Adamson_, Dec 28 2003
%E A091281 a(12)-a(19) from _Charles R Greathouse IV_, Dec 14 2011