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

Author

Al Hakanson (hawkuu(AT)gmail.com), Jan 03 2009

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002163 (decimal expansion of sqrt(5)), A048878.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=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

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 *)

PARI

Vec(x/(1-14*x+44*x^2) + O(x^99)) \\ Altug Alkan, Sep 01 2016

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

Extensions

Extended beyond a(7) by Klaus Brockhaus, Jan 04 2009

Edited by Klaus Brockhaus, Oct 11 2009