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A155001 a(n) = 9*a(n-1) + 72*a(n-2), n > 2; a(0)=1, a(1)=1, a(2)=17. 6
1, 1, 17, 225, 3249, 45441, 642897, 9057825, 127809009, 1802444481, 25424248977, 358594243425, 5057894117169, 71339832581121, 1006226869666257, 14192509772837025, 200180922571503729, 2823489006787799361 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The sequences A155001, A155000, A154999, A154997 and A154996 have a common form: a(0)=a(1)=1, a(2)= 2*b+1, a(n) = (b+1)*(a(n-1) + b*a(n-2)), with b some constant. The generating function of these is (1 - b*x - b^2*x^2)/(1 - (b+1)*x - b*(1+b)*x^2). - R. J. Mathar, Jan 20 2009

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..850

Index entries for linear recurrences with constant coefficients, signature (9,72).

FORMULA

a(n+1) = Sum_{k=0..n} A154929(n,k)*8^(n-k).

a(n) = (1/2)*((9/2 + (3/2)*sqrt(41))^(n-1) + (9/2 - (3/2)*sqrt(41))^(n-1)) + (25/246)*sqrt(41)*((9/2 + (3/2)*sqrt(41))^(n-1) - (9/2 - (3/2)*sqrt(41))^(n-1) + (8/9)*(C(2*n,n) mod 2), n >= 0. - Paolo P. Lava, Jan 20 2009

G.f.: (1 - 8*x - 64*x^2)/(1 - 9*x - 72*x^2). - G. C. Greubel, Apr 20 2021

MAPLE

a[0] := 1: a[1] := 1: a[2] := 17: for n from 3 to 25 do a[n] := 9*a[n-1]+72*a[n-2] end do: seq(a[n], n = 0 .. 17); # Emeric Deutsch, Jan 21 2009

MATHEMATICA

LinearRecurrence[{9, 72}, {1, 1, 17}, 20] (* Harvey P. Dale, Apr 26 2016 *)

PROG

(Magma) I:=[1, 17]; [1] cat [n le 2 select I[n] else 9*(Self(n-1) +8*Self(n-2)): n in [1..30]]; // G. C. Greubel, Apr 20 2021

(Sage)

def A155001_list(prec):

    P.<x> = PowerSeriesRing(ZZ, prec)

    return P( (1-8*x-64*x^2)/(1-9*x-72*x^2) ).list()

A155001_list(30) # G. C. Greubel, Apr 20 2021

CROSSREFS

Cf. A154929, A154996, A154997, A154999, A155000.

Sequence in context: A181380 A081044 A016227 * A012095 A296999 A140842

Adjacent sequences:  A154998 A154999 A155000 * A155002 A155003 A155004

KEYWORD

nonn

AUTHOR

Philippe Deléham, Jan 18 2009

EXTENSIONS

Corrected by Philippe Deléham, Jan 21 2009

Corrected and extended by Emeric Deutsch and R. J. Mathar, Jan 21 2009

STATUS

approved

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Last modified May 11 01:05 EDT 2021. Contains 343784 sequences. (Running on oeis4.)