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URL: https://oeis.org/A153885

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A153885
a(n) = ((8 + sqrt(5))^n - (8 - sqrt(5))^n)/(2*sqrt(5)).
1
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....
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
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