A099920 a(n) = (n+1)*F(n), F(n) = Fibonacci numbers A000045.

0, 2, 3, 8, 15, 30, 56, 104, 189, 340, 605, 1068, 1872, 3262, 5655, 9760, 16779, 28746, 49096, 83620, 142065, 240812, 407353, 687768, 1159200, 1950650, 3277611, 5499704, 9216519, 15426870, 25793240, 43080608, 71884197, 119835652

Offset

0,2

Comments

A Fibonacci-Lucas convolution.

The number of edges in the Lucas cube L_(n+1) [Klavzar]. - R. J. Mathar, Nov 05 2008

Sums of rows of the triangle in A108037. - Reinhard Zumkeller, Oct 07 2012

a(n-1) is the total binary weight of all bimultus bitstrings of length n. A bitstring is bimultus if each of its 1's possess at least one neighboring 1 and each of its 0's possess at least one neighboring 0. - Steven Finch, May 26 2020

References

A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of combinatorial proof, M.A.A. 2003, id. 35.

Links

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

S. Klavzar, On median nature and enumerative properties of Fibonacci-like cubes, Discr. Math. 299 (2005), 145-153.

Franck Ramaharo, A one-variable bracket polynomial for some Turk's head knots, arXiv:1807.05256 [math.CO], 2018.

Steven Finch, Variance of longest run duration in a random bitstring, arXiv:2005.12185 [math.CO], 2020.

Tamás Szakács, Convolution of second order linear recursive sequences. II. Commun. Math. 25, No. 2, 137-148 (2017), remark 2.

Eric Weisstein's World of Mathematics, Edge Count.

Eric Weisstein's World of Mathematics, Lucas Cube Graph.

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

Formula

G.f.: x*(2-x)/(1-x-x^2)^2;

a(n) = Sum_{k=0..n} F(n-k)*(L(k-1) + 0^k).

a(n) = Sum_{k=0..n+1} F(n-k)*binomial(n-k+1, k)*binomial(1, (k+1)/2)*(1-(-1)^k)/2.

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4); a(0)=0, a(1)=2, a(2)=3, a(3)=8. - Harvey P. Dale, Jan 18 2012

a(n) = a(n-1) + a(n-2) + A000032(n-1) (Lucas numbers). - Bob Selcoe, Aug 19 2015

a(n) = 2*A001629(n+1) - A001629(n). - R. J. Mathar, Feb 04 2022

Mathematica

Table[(n + 1) Fibonacci[n], {n, 0, 40}] (* Harvey P. Dale, Jan 18 2012 *)
LinearRecurrence[{2, 1, -2, -1}, {0, 2, 3, 8}, 40] (* Harvey P. Dale, Jan 18 2012 *)
CoefficientList[Series[(2 - x) x/(-1 + x + x^2)^2, {x, 0, 20}], x] (* Eric W. Weisstein, Jul 28 2023 *)

Prog

(Magma) [(n+1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011
(Haskell)
a099920 n = a099920_list !! n
a099920_list = zipWith (*) [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012
(PARI) a(n)=(n+1)*fibonacci(n) \\ Charles R Greathouse IV, Jun 11 2015

Crossrefs

Equals A010049(n) + A001629(n+1).

Cf. A000045, A000032, A045925, A023607.

Sequence in context: A179991 A026698 A099428 * A128022 A011946 A195095

Adjacent sequences: A099917 A099918 A099919 * A099921 A099922 A099923

Keyword

nonn,easy

Author

Paul Barry and Ralf Stephan, Oct 15 2004

Extensions

Entry revised by N. J. A. Sloane, Jan 23 2006. The offset changed, so some of the formulas may now be slightly off.

Status

approved