A023607 - OEIS

%I #75 Dec 30 2025 02:35:59

%S 0,1,4,9,20,40,78,147,272,495,890,1584,2796,4901,8540,14805,25552,

%T 43928,75258,128535,218920,371931,630454,1066464,1800600,3034825,

%U 5106868,8580897,14398412,24129160,40388070,67527579,112786496,188195271,313733810,522562320,869681412

%N a(n) = n * Fibonacci(n+1).

%C Convolution of Fibonacci numbers and Lucas numbers.

%C Central terms of the triangle in A119457 for n > 0. - _Reinhard Zumkeller_, May 20 2006

%C d/dx(1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + ...) = (1 + 4x + 9x^2 + ...). - _Gary W. Adamson_, Jun 27 2009

%C For n > 0: sums of rows of the triangle in A108035. - _Reinhard Zumkeller_, Oct 08 2012

%H Reinhard Zumkeller, <a href="/A023607/b023607.txt">Table of n, a(n) for n = 0..1000</a>

%H Martin Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Griffiths/griffiths16.html">A Restricted Random Walk defined via a Fibonacci Process</a>, Journal of Integer Sequences, Vol. 14 (2011), Article 11.5.4.

%H Milan Janjić, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Janjic/janjic33.html">Hessenberg Matrices and Integer Sequences</a>, J. Int. Seq. 13 (2010), Article 10.7.8, section 3.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1).

%F O.g.f.: x*(2*x+1)/(1-x-x^2)^2. - _Len Smiley_, Dec 11 2001

%F a(n) = n*Sum_{k=0..n} binomial(k,n-k). - _Paul Barry_, Sep 25 2004

%F a(n) = A215082(2n-2) + A215082(2n-1). - _Philippe Deléham_, Aug 03 2012

%F a(n) = Sum_{i=1..n} A000045(i)*A000032(n-i+1). - _Vladimir Kruchinin_, Nov 08 2013

%F 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

%p A023607 := proc(n)

%p     n*combinat[fibonacci](n+1) ;

%p end proc:

%p seq(A023607(n),n=0..10) ; # _R. J. Mathar_, Jul 15 2017

%t Times@@@Thread[{Range[0, 50], Fibonacci[Range[51]]}]  (* _Harvey P. Dale_, Mar 08 2011 *)

%t Table[n*Fibonacci[n + 1], {n, 0, 50}]

%o (Haskell)

%o a023607 n = a023607_list !! n

%o a023607_list = zipWith (*) [0..] $ tail a000045_list

%o -- _Reinhard Zumkeller_, Oct 08 2012

%o (PARI) a(n)=n*fibonacci(n+1) \\ _Charles R Greathouse IV_, Sep 24 2015

%Y First differences of A094584.

%Y Second column of triangle A016095.

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

%K nonn,easy

%O 0,3

%A _Clark Kimberling_

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

%E Name improved by _T. D. Noe_, Mar 08 2011