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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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

a(n) is the number of growing self-avoiding walks with n+3 edges on the grid graph of integer points (x,y) with x >= 0 and y in {0, 1} and with a trapped endpoint. - Jay Pantone, Jul 26 2024

Examples

			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.
		

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
• Jay Pantone, A. R. Klotz, and E. Sullivan, Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height., arXiv:2407.18205 [math.CO], 2024.
• 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).

Crossrefs

Cf. A007492, A066983, A078024, A119282, A014565, A099925, A098600, A374297.

Programs

GAP

List([0..50], n-> Fibonacci(n) + (-1)^n); # G. C. Greubel, Jul 13 2019

Magma

[Fibonacci(n) + (-1)^n: n in [0..50]]; // Vincenzo Librandi, Apr 23 2011

Maple

with(combinat): f := n->fibonacci(n)+(-1)^n; seq(f(n), n=0..40);

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 *)

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

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