A153882 a(n) = ((6 + sqrt(5))^n - (6 - sqrt(5))^n)/(2*sqrt(5)).

1, 12, 113, 984, 8305, 69156, 572417, 4725168, 38957089, 321004860, 2644388561, 21781512072, 179402099473, 1477598319444, 12169714749665, 100231029093216, 825511191878977, 6798972400658028, 55996821859648049, 461193717895377720

Offset: 1

Author

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

Keywords

• nonn

Comments

Fourth binomial transform of A048877.

First differences are in A163146.

lim_{n -> infinity} a(n)/a(n-1) = 6 + sqrt(5) = 8.236067977499789696....

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (12, -31).

Crossrefs

Cf. A002163 (decimal expansion of sqrt(5)), A048877, A163146.

Programs

Magma

Z:= PolynomialRing(Integers()); N:=NumberField(x^2-5); S:=[ ((6+r)^n-(6-r)^n)/(2*r): n in [1..20] ]; [ Integers()!S[j]: j in [1..#S] ]; // Klaus Brockhaus, Jan 04 2009

Magma

I:=[1,12]; [n le 2 select I[n] else 12*Self(n-1)-31*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Sep 01 2016

Mathematica

LinearRecurrence[{12, -31}, {1, 12}, 25] (* or *) Table[((6 + sqrt(5))^n - (6 - sqrt(5))^n)/(2*sqrt(5)) , {n,0,25}] (* G. C. Greubel, Aug 31 2016 *)

Sage

[lucas_number1(n,12,31) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 2009

Formula

From Philippe Deléham, Jan 03 2009: (Start)

a(n) = 12*a(n-1) - 31*a(n-2) for n>1, with a(0)=0, a(1)=1.

G.f.: x/(1 - 12*x + 31*x^2). (End)

a(n) = 12*a(n-1) - 31*a(n-2). - G. C. Greubel, Aug 31 2016

Extensions

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

Edited by Klaus Brockhaus, Oct 11 2009