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

Author

Paul Barry and Ralf Stephan, Oct 15 2004

Keywords

• nonn
• easy

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.
• Tamás Szakács, Linear recursive sequences and factorials, Ph. D. Thesis, Univ. Debrecen (Hungary, 2024). See p. 35.
• 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)

Crossrefs

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

Cf. A000045, A000032, A045925, A023607.

Programs

Haskell

a099920 n = a099920_list !! n
a099920_list = zipWith (*) [1..] a000045_list
-- Reinhard Zumkeller, Oct 07 2012

Magma

[(n+1)*Fibonacci(n): n in [0..60]]; // Vincenzo Librandi, Apr 23 2011

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 *)

PARI

a(n)=(n+1)*fibonacci(n) \\ Charles R Greathouse IV, Jun 11 2015

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

Extensions

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