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

1,2

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).

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

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 *)

Prog

(Magma) Z<x>:= PolynomialRing(Integers()); N<r>:=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
(Sage) [lucas_number1(n, 12, 31) for n in range(1, 21)] # Zerinvary Lajos, Apr 27 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

Crossrefs

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

Sequence in context: A221368 A083767 A285154 * A357650 A322649 A198375

Adjacent sequences: A153879 A153880 A153881 * A153883 A153884 A153885

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