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

0,2

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).

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

Links

Table of n, a(n) for n=0..19.

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

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.

(1/69) {23*15^n - 2*24^[(n+1)/2] + 2*24^[(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]

Example

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

Prog

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

Crossrefs

Cf. A033812.

Sequence in context: A355371 A242945 A243198 * A343009 A368776 A340052

Adjacent sequences: A091278 A091279 A091280 * A091282 A091283 A091284

Keyword

nonn,easy

Author

Gary W. Adamson, Dec 28 2003

Extensions

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

Status

approved