A153885 a(n) = ((8 + sqrt(5))^n - (8 - sqrt(5))^n)/(2*sqrt(5)).

1, 16, 197, 2208, 23705, 249008, 2585533, 26677056, 274286449, 2814636880, 28851289589, 295557057504, 3026686834313, 30989122956272, 317251444075885, 3247664850794112, 33244802412228577, 340304612398804624, 3483430456059387941, 35656915165420734240

Offset

1,2

Comments

Sixth binomial transform of A048879.

lim_{n -> infinity} a(n)/a(n-1) = 8 + sqrt(5) = 10.236067977499789696....

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (16, -59).

Formula

From Philippe Deléham, Jan 03 2009: (Start)

a(n) = 16*a(n-1) - 59*a(n-2) for n>1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 16*x + 59*x^2). (End)

Mathematica

Join[{a=1, b=16}, Table[c=16*b-59*a; a=b; b=c, {n, 40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
LinearRecurrence[{16, -59}, {1, 16}, 25] (* or *) Table[((8 + sqrt(5))^n - (8 - 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:=[ ((8+r)^n-(8-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; # Klaus Brockhaus, Jan 04 2009
(Magma) I:=[1, 16]; [n le 2 select I[n] else 16*Self(n-1)-59*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 01 2016

Crossrefs

Cf. A002163 (decimal expansion of sqrt(5)), A048879.

Sequence in context: A103721 A144844 A093060 * A016226 A332854 A154240

Adjacent sequences: A153882 A153883 A153884 * A153886 A153887 A153888

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