A153599 a(n) = ((8 + sqrt(3))^n - (8 - sqrt(3))^n)/(2*sqrt(3)).

1, 16, 195, 2144, 22409, 227760, 2277211, 22542016, 221762385, 2173135184, 21242657459, 207321273120, 2021338264921, 19694814578416, 191815399094475, 1867662696228224, 18181863794888609, 176982396248296080, 1722624648484532131, 16766068204606453216

Offset

1,2

Comments

lim_{n -> infinity} a(n)/a(n-1) = 8 + sqrt(3) = 9.73205080756887729....

Links

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

Index entries for linear recurrences with constant coefficients, signature (16, -61).

Formula

G.f.: x/(1 - 16*x + 61*x^2). - Klaus Brockhaus, Dec 31 2008, (corrected Oct 11 2009)

a(n) = 16*a(n-1) - 61*a(n-2) for n>1; a(0)=0, a(1)=1. - Philippe Deléham, Jan 01 2009

Mathematica

Join[{a=1, b=16}, Table[c=16*b-61*a; a=b; b=c, {n, 40}]] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2011*)
Simplify/@Table[c=Sqrt[3]; ((8+c)^n-(8-c)^n)/(2c), {n, 20}] (* or *) LinearRecurrence[{16, -61}, {1, 16}, 20] (* Harvey P. Dale, Sep 24 2012 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-3); S:=[ ((8+r)^n-(8-r)^n)/(2*r): n in [1..18] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Dec 31 2008
(Magma)  I:=[1, 16]; [n le 2 select I[n] else 16*Self(n-1)-61*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 23 2016

Crossrefs

Cf. A002194 (decimal expansion of sqrt(3)).

Sequence in context: A292785 A081185 A159517 * A016280 A281946 A077363

Adjacent sequences: A153596 A153597 A153598 * A153600 A153601 A153602

Keyword

nonn

Author

Al Hakanson (hawkuu(AT)gmail.com), Dec 29 2008

Extensions

Extended beyond a(7) by Klaus Brockhaus, Dec 31 2008

Edited by Klaus Brockhaus, Oct 11 2009

Status

approved