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, 27, 649, 15603, 375121, 9018507, 216819289, 5212681443, 125321173921, 3012920855547, 72435421707049, 1741463041824723, 41867548425500401, 1006562625253834347, 24199370554517524729, 581791455933674427843, 13987194312962703792961, 336274454967038565458907
Offset: 1
Examples
a(4)=15603 because M^4 = [15603,26590,42193 / 56642,96527,153169 / 82078,139874,221952]. Alternatively, a(4) = 24*649+27 = 15603.
Links
- Colin Barker, Table of n, a(n) for n = 1..700
- Tanya Khovanova, Recursive Sequences
- Index entries for linear recurrences with constant coefficients, signature (24,1).
Programs
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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);
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PARI
Vec(x*(1 + 3*x) / (1 - 24*x - x^2) + O(x^30)) \\ Colin Barker, Mar 02 2017
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
Extensions
Corrected by T. D. Noe, Nov 07 2006
Edited by N. J. A. Sloane, Dec 16 2006