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A124610 a(n) = 5*a(n-1) + 2*a(n-2), n>1; a(0) = a(1) = 1. 4
1, 1, 7, 37, 199, 1069, 5743, 30853, 165751, 890461, 4783807, 25699957, 138067399, 741736909, 3984819343, 21407570533, 115007491351, 617852597821, 3319277971807, 17832095054677, 95799031216999, 514659346194349 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Top left element of powers of the matrix [1,2;3,4].

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (5, 2).

FORMULA

a(n)/a(n-1) tends to (sqrt(33) + 5)/2 = 5.37228132... - Gary W. Adamson, Mar 03 2008

a(n) = -(sqrt(33)/22)*((5+sqrt(33))/2)^n + (1/2)*((5-sqrt(33))/2)^n + (sqrt(33)/22)*((5-sqrt(33))/2)^n + (1/2)*((5+sqrt(33))/2)^n, with n>=0. - Paolo P. Lava, Jul 07 2008

G.f.: (1 - 4*x)/(1 - 5*x - 2*x^2). - G. C. Greubel, Oct 23 2019

EXAMPLE

a(5) = 1069 because [1,2;3,4]^5 = [1069,1558;2337,3406]

MAPLE

seq(coeff(series((1-4*x)/(1-5*x-2*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 23 2019

MATHEMATICA

Table[MatrixPower[{{1, 2}, {3, 4}}, n][[1]][[1]], {n, 0, 30}]

Transpose[NestList[Flatten[{Rest[#], ListCorrelate[{2, 5}, #]}]&, {1, 1}, 40]][[1]]  (* Harvey P. Dale, Mar 23 2011 *)

LinearRecurrence[{5, 2}, {1, 1}, 30] (* Harvey P. Dale, Jan 01 2014 *)

PROG

(PARI) Vec((1-4*x)/(1-5*x-2*x^2) +O('x^30)) \\ G. C. Greubel, Oct 23 2019

(MAGMA) R<x>:=PowerSeriesRing(Integers(), 30); Coefficients(R!( (1-4*x)/(1-5*x-2*x^2) )); // G. C. Greubel, Oct 23 2019

(MAGMA) [n le 2 select 1 else 5*Self(n-1) + 2*Self(n-2):n in [1..22]]; // Marius A. Burtea, Oct 24 2019

(Sage)

def A124610_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( (1-4*x)/(1-5*x-2*x^2) ).list()

A124610_list(30) # G. C. Greubel, Oct 23 2019

(GAP) a:=[1, 1];; for n in [3..30] do a[n]:=5*a[n-1]+2*a[n-2]; od; a; # G. C. Greubel, Oct 23 2019

CROSSREFS

Cf. A100638.

Sequence in context: A287808 A117130 A002807 * A002683 A319013 A126475

Adjacent sequences:  A124607 A124608 A124609 * A124611 A124612 A124613

KEYWORD

easy,nonn

AUTHOR

Fredrik Johansson, Dec 20 2006

EXTENSIONS

Recurrence from Gary W. Adamson, Mar 03 2008

STATUS

approved

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Last modified November 30 05:31 EST 2020. Contains 338781 sequences. (Running on oeis4.)