A023607 a(n) = n * Fibonacci(n+1).

0, 1, 4, 9, 20, 40, 78, 147, 272, 495, 890, 1584, 2796, 4901, 8540, 14805, 25552, 43928, 75258, 128535, 218920, 371931, 630454, 1066464, 1800600, 3034825, 5106868, 8580897, 14398412, 24129160, 40388070, 67527579, 112786496, 188195271, 313733810, 522562320, 869681412

Offset

0,3

Comments

Convolution of Fibonacci numbers and Lucas numbers.

Central terms of the triangle in A119457 for n > 0. - Reinhard Zumkeller, May 20 2006

d/dx(1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) = (1 + 4x + 9x^2 + ...). - Gary W. Adamson, Jun 27 2009

For n > 0: sums of rows of the triangle in A108035. - Reinhard Zumkeller, Oct 08 2012

Links

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

Martin Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.4.

Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010), Article 10.7.8, section 3.

Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Formula

O.g.f.: x*(2*x+1)/(1-x-x^2)^2. - Len Smiley, Dec 11 2001

a(n) = n*Sum_{k=0..n} binomial(k,n-k). - Paul Barry, Sep 25 2004

a(n) = A215082(2n-2) + A215082(2n-1). - Philippe Deléham, Aug 03 2012

a(n) = Sum_{i=1..n} A000045(i)*A000032(n-i+1). - Vladimir Kruchinin, Nov 08 2013

E.g.f.: (x/5) * exp(x/2) * (5*cosh(sqrt(5)*x/2) + 3*sqrt(5)*sinh(sqrt(5)*x/2)). - Amiram Eldar, Dec 30 2025

Maple

A023607 := proc(n)
    n*combinat[fibonacci](n+1) ;
end proc:
seq(A023607(n), n=0..10) ; # R. J. Mathar, Jul 15 2017

Mathematica

Times@@@Thread[{Range[0, 50], Fibonacci[Range[51]]}]  (* Harvey P. Dale, Mar 08 2011 *)
Table[n*Fibonacci[n + 1], {n, 0, 50}]

Prog

(Haskell)
a023607 n = a023607_list !! n
a023607_list = zipWith (*) [0..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 08 2012
(PARI) a(n)=n*fibonacci(n+1) \\ Charles R Greathouse IV, Sep 24 2015

Crossrefs

First differences of A094584.

Second column of triangle A016095.

Cf. A000032, A000045, A104796, A108035, A119457, A215082.

Sequence in context: A049748 A268235 A192956 * A117074 A072934 A084639

Adjacent sequences: A023604 A023605 A023606 * A023608 A023609 A023610

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

Simpler description from Samuel Lachterman (slachterman(AT)fuse.net), Sep 19 2003

Name improved by T. D. Noe, Mar 08 2011

Status

approved