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
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
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