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

1, 5, 91, 1125, 17259, 253125, 3806091, 56953125, 854518059, 12814453125, 192222105291, 2883251953125, 43248906698859, 648731689453125, 9730978399444491, 145964630126953125, 2189469525287839659, 32842041778564453125, 492630628439671823691, 7389459400177001953125

Offset: 0

Author

Gary W. Adamson, Dec 28 2003

Keywords

• nonn
• easy

Comments

a(n)/a(n-1) tends to 15, the "Magic Number" of the Lo-Shu Magic Square.

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).

Examples

			a(2) = 91 since M^2 = [91, 67, 67 / 67, 91, 67 / 67, 67, 91].
		

Links

• Index entries for linear recurrences with constant coefficients, signature (15,24,-360).

Crossrefs

Cf. A033812.

Programs

PARI

a(n)=([8,1,6;3,5,7;4,9,2]^n)[2,2] \\ Charles R Greathouse IV, Dec 14 2011

Formula

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.

a(2*k+1) = 5*15^(2*k). E.g. a(5) = 253125 = 5*15^4.

a(n) = (1/69)*(23*15^n - 2*24^floor((n+1)/2) + 2*24^floor((n+2)/2)). - Ralf Stephan, Dec 02 2004

G.f.: -(8*x^2+10*x-1) / ((15*x-1)*(24*x^2-1)). - Colin Barker, Dec 10 2012

Extensions

a(12)-a(19) from Charles R Greathouse IV, Dec 14 2011