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URL: https://oeis.org/A153884

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A153884
a(n) = ((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)).
1
1, 14, 152, 1512, 14480, 136192, 1269568, 11781504, 109080064, 1008734720, 9322763264, 86134358016, 795679428608, 7349600247808, 67884508610560, 627000709644288, 5791091556155392, 53487250561826816, 494013479394738176
OFFSET
1,2
COMMENTS
Fifth binomial transform of A048878.
lim_{n -> infinity} a(n)/a(n-1) = 7 + sqrt(5) = 9.236067977499789696....
LINKS
S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.
FORMULA
From Philippe Deléham, Jan 03 2009: (Start)
a(n) = 14*a(n-1) - 44*a(n-2) for n>1, with a(0)=0, a(1)=1.
G.f.: x/(1 - 14*x + 44*x^2). (End)
E.g.f.: sinh(sqrt(5)*x)*exp(7*x)/sqrt(5). - Ilya Gutkovskiy, Sep 01 2016
MATHEMATICA
Join[{a=1, b=14}, Table[c=14*b-44*a; a=b; b=c, {n, 60}]] (* Vladimir Joseph Stephan Orlovsky, Feb 01 2011 *)
LinearRecurrence[{14, -44}, {1, 14}, 25] (* or *) Table[((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)) , {n, 0, 25}] (* G. C. Greubel, Aug 31 2016 *)
PROG
(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=NumberField(x^2-5); S:=[ ((7+r)^n-(7-r)^n)/(2*r): n in [1..19] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 04 2009
(Magma) I:=[1, 14]; [n le 2 select I[n] else 14*Self(n-1)-44*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 01 2016
(PARI) Vec(x/(1-14*x+44*x^2) + O(x^99)) \\ Altug Alkan, Sep 01 2016
CROSSREFS
Cf. A002163 (decimal expansion of sqrt(5)), A048878.
Sequence in context: A037960 A222677 A016163 * A154239 A016215 A329711
KEYWORD
nonn,easy
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