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URL: https://oeis.org/A153886

⇱ A153886 - OEIS

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A153886
a(n) = ((9 + sqrt(5))^n - (9 - sqrt(5))^n)/(2*sqrt(5)).
1
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....
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 A385029 A154250
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