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A122883 The (1,3)-entry in the 3 X 3 matrix M^n, where M = [1,1,1 / 4,2,1 / 9,3,1]. 4
1, 3, 23, 123, 739, 4263, 24935, 145155, 846379, 4932351, 28749263, 167560155, 976617811, 5692134423, 33176213303, 193365096243, 1127014462459, 6568721481903, 38285314822175, 223143166664715, 1300573686738979, 7580298950623431, 44181220023293063 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (4,11,-2)

FORMULA

a(n) = 4*a(n-1) + 11*a(n-2) - 2*a(n-3) (derived from the minimal polynomial of the matrix M).

G.f.: -x*(-1+x)/((2*x+1)*(1-6*x+x^2)). a(n) = -3*(-2)^n/17+(3*A001109(n+1)-7*A001109(n))/17. - R. J. Mathar, Aug 12 2009

a(n) =  (-3*(-1)^n*2^(2+n) - (3-2*sqrt(2))^n*(-6+sqrt(2)) + (6+sqrt(2))*(3+2*sqrt(2))^n) / 68. - Colin Barker, Mar 02 2017

EXAMPLE

a(6) = 4263 because M^3 = [13742,6930,4263; 25053,12671,7819; 41034,20790,12853]; alternatively, a(6) = 4*a(5) + 11*a(4) - 2*a(3) = 4*729+11*123-2*23 = 4263.

MAPLE

with(linalg): M[1]:=matrix(3, 3, [1, 1, 1, 4, 2, 1, 9, 3, 1]): for n from 2 to 25 do M[n]:=multiply(M[1], M[n-1]) od: seq(M[n][1, 3], n=1..25);

a[1]:=1:a[2]:=3:a[3]:=23:for n from 4 to 25 do a[n]:=4*a[n-1]+11*a[n-2]-2*a[n-3] od: seq(a[n], n=1..25);

PROG

(PARI) Vec(x*(1 - x) / ((1 + 2*x)*(1 - 6*x + x^2)) + O(x^30)) \\ Colin Barker, Mar 02 2017

CROSSREFS

Cf. A122884, A122885, A122886.

Sequence in context: A226611 A081628 A267656 * A196424 A091055 A154648

Adjacent sequences:  A122880 A122881 A122882 * A122884 A122885 A122886

KEYWORD

nonn,easy

AUTHOR

Gary W. Adamson and Roger L. Bagula, Sep 17 2006

EXTENSIONS

Edited by N. J. A. Sloane, Dec 04 2006

STATUS

approved

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Last modified August 4 09:55 EDT 2021. Contains 346446 sequences. (Running on oeis4.)