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A095898
The (1,1)-term of the 3 X 3 matrix M^n, where M = [1,2,3 / 4,7,11 / 6,10,16].
1
1, 27, 649, 15603, 375121, 9018507, 216819289, 5212681443, 125321173921, 3012920855547, 72435421707049, 1741463041824723, 41867548425500401, 1006562625253834347, 24199370554517524729, 581791455933674427843, 13987194312962703792961, 336274454967038565458907
OFFSET
1,2
FORMULA
a(n) = 24*a(n-1) + a(n-2) for n>=3; a(1)=1, a(2)=27 (follows from the minimal polynomial of the matrix M).
G.f.: (x+3*x^2) / (1-24*x-x^2). - Philippe Deléham, Nov 21 2008
a(n) = (-12 - sqrt(145))^(-n)*(87+7*sqrt(145) + (-289-24*sqrt(145))^n*(87-7*sqrt(145))) / 58. - Colin Barker, Mar 02 2017
EXAMPLE
a(4)=15603 because M^4 = [15603,26590,42193 / 56642,96527,153169 / 82078,139874,221952]. Alternatively, a(4) = 24*649+27 = 15603.
MAPLE
a[1]:=1: a[2]:=27: for n from 3 to 18 do a[n]:=24*a[n-1]+a[n-2] od: seq(a[n], n=1..18);
PROG
(PARI) Vec(x*(1 + 3*x) / (1 - 24*x - x^2) + O(x^30)) \\ Colin Barker, Mar 02 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Gary W. Adamson, Jun 12 2004
EXTENSIONS
Corrected by T. D. Noe, Nov 07 2006
Edited by N. J. A. Sloane, Dec 16 2006
STATUS
approved