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A081569 Fourth binomial transform of F(n+1). 8
1, 5, 26, 139, 757, 4172, 23165, 129217, 722818, 4050239, 22718609, 127512940, 715962889, 4020920141, 22584986378, 126867394723, 712691811325, 4003745802188, 22492567804517, 126361939999081, 709898671705906 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Binomial transform of A081568.

Case k=4 of family of recurrences a(n) = (2k+1)*a(n-1) - A028387(k-1)*a(n-2), a(0)=1, a(1)=k+1.

a(n) = 5^n * a(n;1/5) = Sum_{k=0..n} binomial(n,k) * (-1)^k * F(k-1) * 5^(n-k), which implies also Deléham's formula given below and where a(n;d), n=0,1,..., d, denote the delta-Fibonacci numbers defined in comments to A000045 (see also Witula's et al. papers). - Roman Witula, Jul 12 2012

REFERENCES

D. Chmiela, K. Kaczmarek, R. Witula, Binomials Transformation Formulae of Scaled Fibonacci Numbers, (submitted to Fibonacci Quart. 2012).

LINKS

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

R. Witula, Damian Slota, delta-Fibonacci numbers, Appl. Anal. Discr. Math 3 (2009) 310-329, MR2555042

Index entries for linear recurrences with constant coefficients, signature (9,-19)

FORMULA

a(n) = 9*a(n-1) - 19*a(n-2), a(0)=1, a(1)=5.

a(n) = (1/2-sqrt(5)/10)*(9/2-sqrt(5)/2)^n + (sqrt(5)/10+1/2)*(sqrt(5)/2+9/2)^n.

G.f.: (1-4*x)/(1-9*x+19*x^2).

a(n) = sum(k=0..n, A094441(n,k)*4^k ). - Philippe Deléham, Dec 14 2009

a(n) = A081574(n) - 4*A081574(n-1). - R. J. Mathar, Jul 19 2012

MATHEMATICA

CoefficientList[Series[(1 - 4 x) / (1 - 9 x + 19 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, Aug 09 2013 *)

PROG

(MAGMA) I:=[1, 5]; [n le 2 select I[n] else 9*Self(n-1)-19*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Aug 09 2013

(PARI) Vec((1-4*x)/(1-9*x+19*x^2) + O(x^100)) \\ Altug Alkan, Dec 10 2015

CROSSREFS

Cf. A000045.

Sequence in context: A161731 A049607 A035029 * A005573 A081911 A081187

Adjacent sequences:  A081566 A081567 A081568 * A081570 A081571 A081572

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Mar 22 2003

STATUS

approved

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Last modified March 22 01:06 EDT 2019. Contains 321406 sequences. (Running on oeis4.)