A008346 a(n) = Fibonacci(n) + (-1)^n.

1, 0, 2, 1, 4, 4, 9, 12, 22, 33, 56, 88, 145, 232, 378, 609, 988, 1596, 2585, 4180, 6766, 10945, 17712, 28656, 46369, 75024, 121394, 196417, 317812, 514228, 832041, 1346268, 2178310, 3524577, 5702888, 9227464, 14930353, 24157816, 39088170, 63245985, 102334156

Offset

0,3

Comments

Diagonal sums of A059260. - Paul Barry, Oct 25 2004

The absolute value of the Euler characteristic of the Boolean complex of the Coxeter group A_n. - Bridget Tenner, Jun 04 2008

a(n) is the number of compositions (ordered partitions) of n into two sorts of 2's and one sort of 3's. Example: the a(5)=4 compositions of 5 are 2+3, 2'+3, 3+2 and 3+2'. - Bob Selcoe, Jun 21 2013

Let r = 0.70980344286129... denote the rabbit constant A014565. The sequence 2^a(n) gives the simple continued fraction expansion of the constant r/2 = 0.35490172143064565732 ... = 1/(2^1 + 1/(2^0 + 1/(2^2 + 1/(2^1 + 1/(2^4 + 1/(2^4 + 1/(2^9 + 1/(2^12 + ... )))))))). Cf. A099925. - Peter Bala, Nov 06 2013

a(n) is the top left entry of the n-th power of the 3 X 3 matrix [0, 1, 1; 1, 0, 1; 1, 0, 0] or of the 3 X 3 matrix [0, 1, 1; 1, 0, 0; 1, 1, 0]. - R. J. Mathar, Feb 03 2014

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

G. Bilgici, Generalized order-k Pell-Padovan-like numbers by matrix methods, Pure and Applied Mathematics Journal, 2013; 2(6): 174-178.

Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, Predators and altruists arriving on jammed Riviera, arXiv:2401.01225 [math.CO], 2024. See p. 14.

N. Gogin and A. Mylläri, Padovan-like sequences and Bell polynomials, Proceedings of Applications of Computer Algebra ACA, 2013.

Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 445

K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system, arXiv:0806.0906 [math.CO], 2008-2009.

Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Formula

G.f.: 1/(1 - 2*x^2 - x^3).

a(n) = 2*a(n-2) + a(n-3).

a(n) = Sum_{k=0..floor(n/2)} Sum_{j=0..n-k} (-1)^(n-k-j)binomial(j, k). Diagonal sums of A059260. - Paul Barry, Sep 23 2004

From Paul Barry, Oct 04 2004: (Start)

a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^(3k-n).

a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^k(1/2)^(n-2k). (End)

From Paul Barry, Oct 25 2004: (Start)

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

a(n) = Sum_{k=0..n} binomial(n-k-1, k). (End)

a(n) = |1 + (-1)^(n-1)*Fibonacci(n-1)|. - Bridget Tenner, Jun 04 2008

a(n) = A000045(n) + A033999(n). - Michel Marcus, Nov 14 2013

a(n) = Fibonacci(n+1) - a(n-1), with a(0) = 1. - Franklin T. Adams-Watters, Mar 26 2014

a(n) = b(n+1) where b(n) = b(n-1) + b(n-2) + (-1)^(n+1), b(0) = 0, b(1) = 1. See also A098600. - Richard R. Forberg, Aug 30 2014

a(n) = b(n+2) where b(n) = Sum_{k=1..n} b(n-k)*A000931(k+1), b(0) = 1. - J. Conrad, Apr 19 2017

a(n) = Sum_{j=n+1..2*n+1} F(j) mod Sum_{j=0..n} F(j) for n > 2 and F(j)=A000045(j). - Art Baker, Jan 20 2019

Example

The Boolean complex of Coxeter group A_4 is homotopy equivalent to the wedge of 2 spheres S^3, which has Euler characteristic 1 - 2 = -1.

Maple

with(combinat): f := n->fibonacci(n)+(-1)^n;

Mathematica

Table[Fibonacci[n]+(-1)^n, {n, 0, 50}] (* Vladimir Joseph Stephan Orlovsky, Jul 22 2008 *)
CoefficientList[Series[1/(1-2x^2-x^3), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 10 2013 *)
LinearRecurrence[{0, 2, 1}, {1, 0, 2}, 51] (* Ray Chandler, Sep 08 2015 *)

Prog

(Magma) [Fibonacci(n) + (-1)^n: n in [0..50]]; // Vincenzo Librandi, Apr 23 2011
(PARI) a(n)=fibonacci(n)+(-1)^n \\ Charles R Greathouse IV, Feb 03 2014
(Sage) [fibonacci(n)+(-1)^n for n in (0..50)] # G. C. Greubel, Jul 13 2019
(GAP) List([0..50], n-> Fibonacci(n) + (-1)^n) # G. C. Greubel, Jul 13 2019

Crossrefs

Cf. A007492, A066983, A078024, A119282, A014565, A099925, A098600.

Sequence in context: A099932 A175000 A355020 * A119282 A241513 A095293

Adjacent sequences: A008343 A008344 A008345 * A008347 A008348 A008349

Keyword

nonn,easy

Author

N. J. A. Sloane

Status

approved