A099925 -id:A099925 - cp's OEIS frontend

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

N. J. A. Sloane

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

			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.

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).

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

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

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

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

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 *)

a(n)=fibonacci(n)+(-1)^n \\ Charles R Greathouse IV, Feb 03 2014

[fibonacci(n)+(-1)^n for n in (0..50)] # G. C. Greubel, Jul 13 2019

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

A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

Original entry on oeis.org

1, 2, 2, 5, 6, 12, 17, 30, 46, 77, 122, 200, 321, 522, 842, 1365, 2206, 3572, 5777, 9350, 15126, 24477, 39602, 64080, 103681, 167762, 271442, 439205, 710646, 1149852, 1860497, 3010350, 4870846, 7881197, 12752042, 20633240, 33385281, 54018522

Offset: 0

Paul Barry, Sep 17 2004

Row sums of A098599.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Aleksandar Petojević, Marjana Gorjanac Ranitović, Dragan Rastovac, and Milinko Mandić, The Golden Ratio, Factorials, and the Lambert W Function, Journal of Integer Sequences, Vol. 27 (2024), Article 24.5.7. See p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,2,1).

Cf. A000032, A000035, A000045, A001350, A008346.
Cf. A020878, A098599, A099925, A181716, A182143.
First differences of A014217 and A062724.

[Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n: n in [0..50]]; // Vincenzo Librandi, Aug 31 2014

Table[-(-1)^n + LucasL[n], {n, 0, 39}] (* Alonso del Arte, Aug 30 2014 *)
Table[Fibonacci[n - 1] + Fibonacci[n + 1] - (-1)^n, {n, 0, 40}] (* Vincenzo Librandi, Aug 31 2014 *)
CoefficientList[ Series[-(1 + 2x)/(-1 + 2x^2 + x^3), {x, 0, 40}], x] (* or *)
LinearRecurrence[{0, 2, 1}, {1, 2, 2}, 40] (* Robert G. Wilson v, Mar 09 2018 *)

a(n)=fibonacci(n-1) + fibonacci(n+1) - (-1)^n; \\ Joerg Arndt, Oct 18 2014

Vec((1+2*x)/((1+x)*(1-x-x^2)) + O(x^30)) \\ Colin Barker, Jun 03 2016

[lucas_number2(n,1,-1) -(-1)^n for n in range(51)] # G. C. Greubel, Mar 26 2024

G.f.: (1+2*x) / ((1+x)*(1-x-x^2)).
a(n) = Sum_{k = 0..n} binomial(k, n-k) + binomial(k-1, n-k-1).
a(n) = A020878(n) - 1 = A001350(n) + 1.
a(n) = Lucas(n) - (-1)^n. - Paul Barry, Dec 01 2004
a(n) = A181716(n+1). - Richard R. Forberg, Aug 30 2014
a(n) = [x^n] ( (1 + x + sqrt(1 + 6*x + 5*x^2))/2 )^n. exp( Sum_{n >= 1} a(n)*x^n/n ) = Sum_{n >= 0} Fibonacci(n+2)*x^n. Cf. A182143. - Peter Bala, Jun 29 2015
From Colin Barker, Jun 03 2016: (Start)
a(n) = (-(-1)^n + ((1/2)*(1-sqrt(5)))^n + ((1/2)*(1+sqrt(5)))^n).
a(n) = 2*a(n-2) + a(n-3) for n > 2. (End)
E.g.f.: (2*exp(3*x/2)*cosh(sqrt(5)*x/2) - 1)*exp(-x). - Ilya Gutkovskiy, Jun 03 2016
a(n) = A014217(n) + A000035(n). - Paul Curtz, Jul 27 2023

cp's OEIS Frontend

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

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A098600 a(n) = Fibonacci(n-1) + Fibonacci(n+1) - (-1)^n.

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Magma

Mathematica

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