A081575 Fifth binomial transform of Fibonacci numbers F(n).

0, 1, 11, 92, 693, 4955, 34408, 234793, 1584891, 10624804, 70911005, 471901739, 3134499984, 20794349393, 137837343787, 913174649260, 6047638172037, 40041955063867, 265079998713464, 1754663288995961, 11613976216265115

Offset

0,3

Comments

Binomial transform of A081574.

Case k=5 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=0, a(1)=1.

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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 (11,-29).

Formula

a(n) = 11*a(n-1) - 29*a(n-2), a(0)=0, a(1)=1.

a(n) = ((sqrt(5)/2 + 11/2)^n - (11/2 - sqrt(5)/2)^n)/sqrt(5).

G.f.: x/(1 - 11*x + 29*x^2). - adapted by Vincenzo Librandi, Aug 09 2013

E.g.f.: 2*exp(11*x/2)*sinh(sqrt(5)*x/2)/sqrt(5). - Ilya Gutkovskiy, Aug 12 2017

Maple

seq(coeff(series(x/(1-11*x+29*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Aug 13 2019

Mathematica

LinearRecurrence[{11, -29}, {0, 1}, 30] (* Vladimir Joseph Stephan Orlovsky, Jan 31 2011; modified by G. C. Greubel, Aug 13 2019 *)
CoefficientList[Series[x/(1 -11x +29x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

Prog

(Sage) [lucas_number1(n, 11, 29) for n in range(0, 21)] # Zerinvary Lajos, Apr 27 2009
(Magma) [n le 2 select (n-1) else 11*Self(n-1)-29*Self(n-2): n in [1..25]]; // Vincenzo Librandi, Aug 09 2013
(PARI) my(x='x+O('x^30)); Vec(x/(1-11*x+29*x^2)) \\ G. C. Greubel, Aug 13 2019
(GAP) a:=[0, 1];; for n in [3..30] do a[n]:=11*a[n-1]-29*a[n-2]; od; a; # G. C. Greubel, Aug 13 2019

Crossrefs

Cf. A000045, A081570.

Sequence in context: A044262 A044643 A025507 * A016150 A115203 A164547

Adjacent sequences: A081572 A081573 A081574 * A081576 A081577 A081578

Keyword

easy,nonn

Author

Paul Barry, Mar 24 2003

Status

approved