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

1,2

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).

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

Cf. A083412, A035513, A003622, A001950, A000201.

Sequence in context: A060603 A116988 A113364 * A014914 A157461 A342037

Adjacent sequences: A095895 A095896 A095897 * A095899 A095900 A095901

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