A153884 a(n) = ((7 + sqrt(5))^n - (7 - sqrt(5))^n)/(2*sqrt(5)).

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

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

S. Falcon, Iterated Binomial Transforms of the k-Fibonacci Sequence, British Journal of Mathematics & Computer Science, 4 (22): 2014.

Index entries for linear recurrences with constant coefficients, signature (14,-44).

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

Adjacent sequences: A153881 A153882 A153883 * A153885 A153886 A153887

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