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
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