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

Offset: 0

Author

Clark Kimberling

Keywords

• nonn
• easy

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
• M. Griffiths, A Restricted Random Walk defined via a Fibonacci Process, Journal of Integer Sequences, Vol. 14 (2011), #11.5.4.
• Milan Janjić, Hessenberg Matrices and Integer Sequences, J. Int. Seq. 13 (2010) # 10.7.8, section 3.
• Index entries for linear recurrences with constant coefficients, signature (2,1,-2,-1).

Crossrefs

First differences of A094584.

Second column of triangle A016095.

Cf. A000045, A104796.

Programs

Haskell

a023607 n = a023607_list !! n
a023607_list = zipWith (*) [0..] $ tail a000045_list
-- Reinhard Zumkeller, Oct 08 2012

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}]

PARI

a(n)=n*fibonacci(n+1) \\ Charles R Greathouse IV, Sep 24 2015

Formula

O.g.f.: x(2x+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

Extensions

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

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