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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
OFFSET
1,2
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);
MATHEMATICA
LinearRecurrence[{4, 11, -2}, {1, 3, 23}, 25] (* Paolo Xausa, Jul 19 2024 *)
PROG
(PARI) Vec(x*(1 - x) / ((1 + 2*x)*(1 - 6*x + x^2)) + O(x^30)) \\ Colin Barker, Mar 02 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Dec 04 2006
STATUS
approved