A153886 a(n) = ((9 + sqrt(5))^n - (9 - sqrt(5))^n)/(2*sqrt(5)).

1, 18, 248, 3096, 36880, 428544, 4910912, 55827072, 631657984, 7126986240, 80279745536, 903384465408, 10159659716608, 114216655527936, 1283765661040640, 14427316078608384, 162125499175862272, 1821782963191283712, 20470555400077574144

Offset

1,2

Comments

lim_{n -> infinity} a(n)/a(n-1) = 9 + sqrt(5) = 11.236067977499789696....

Links

G. C. Greubel, Table of n, a(n) for n = 1..950

Index entries for linear recurrences with constant coefficients, signature (18, -76).

Formula

From Philippe Deléham and Klaus Brockhaus, Jan 03 2009: (Start)

a(n) = 18*a(n-1) - 76*a(n-2) for n>1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 18*x + 76*x^2). (End)

E.g.f.: sinh(sqrt(5)*x)*exp(9*x)/sqrt(5). - Ilya Gutkovskiy, Sep 01 2016

Mathematica

Join[{a=1, b=18}, Table[c=18*b-76*a; a=b; b=c, {n, 40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 09 2011 *)
LinearRecurrence[{18, -76}, {1, 18}, 20] (* Harvey P. Dale, Jun 06 2011 *)
Table[Simplify[((9 + Sqrt[5])^n - (9 - Sqrt[5])^n)/(2 Sqrt[5])], {n, 1, 25}] (* G. C. Greubel, Aug 31 2016 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((9+r)^n-(9-r)^n)/(2*r): n in [1..17] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 04 2009
(Magma)  I:=[1, 18]; [n le 2 select I[n] else 18*Self(n-1)-76*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 01 2016
(PARI) Vec(x/(1-18*x+76*x^2) + O(x^99)) \\ Altug Alkan, Sep 01 2016

Crossrefs

Cf. A002163 (decimal expansion of sqrt(5)).

Sequence in context: A153600 A016183 A016239 * A154241 A154250 A154350

Adjacent sequences: A153883 A153884 A153885 * A153887 A153888 A153889

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 04 2009

Edited by Klaus Brockhaus, Oct 11 2009

Status

approved