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%I #136 Jul 27 2024 04:00:32
%S 1,0,2,1,4,4,9,12,22,33,56,88,145,232,378,609,988,1596,2585,4180,6766,
%T 10945,17712,28656,46369,75024,121394,196417,317812,514228,832041,
%U 1346268,2178310,3524577,5702888,9227464,14930353,24157816,39088170,63245985,102334156
%N a(n) = Fibonacci(n) + (-1)^n.
%C Diagonal sums of A059260. - _Paul Barry_, Oct 25 2004
%C The absolute value of the Euler characteristic of the Boolean complex of the Coxeter group A_n. - _Bridget Tenner_, Jun 04 2008
%C 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
%C 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
%C 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
%C 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
%H Vincenzo Librandi, <a href="/A008346/b008346.txt">Table of n, a(n) for n = 0..1000</a>
%H G. Bilgici, <a href="http://dx.doi.org/10.11648/j.pamj.20130206.11">Generalized order-k Pell-Padovan-like numbers by matrix methods</a>, Pure and Applied Mathematics Journal, 2013; 2(6): 174-178.
%H Tomislav Došlić, Mate Puljiz, Stjepan Šebek, and Josip Žubrinić, <a href="https://arxiv.org/abs/2401.01225">Predators and altruists arriving on jammed Riviera</a>, arXiv:2401.01225 [math.CO], 2024. See p. 14.
%H N. Gogin and A. Mylläri, <a href="http://www.aca2013.uma.es/Proceedings.pdf#page=184">Padovan-like sequences and Bell polynomials</a>, Proceedings of Applications of Computer Algebra ACA, 2013.
%H Jia Huang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Huang/huang8.html">Partially Palindromic Compositions</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 13.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=445">Encyclopedia of Combinatorial Structures 445</a>
%H Jay Pantone, A. R. Klotz, and E. Sullivan, <a href="https://arxiv.org/abs/2407.18205">Exactly-solvable self-trapping lattice walks. II. Lattices of arbitrary height.</a>, arXiv:2407.18205 [math.CO], 2024.
%H K. Ragnarsson and B. E. Tenner, <a href="http://arxiv.org/abs/0806.0906">Homotopy type of the Boolean complex of a Coxeter system</a>, arXiv:0806.0906 [math.CO], 2008-2009.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,1).
%F G.f.: 1/(1 - 2*x^2 - x^3).
%F a(n) = 2*a(n-2) + a(n-3).
%F 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
%F From _Paul Barry_, Oct 04 2004: (Start)
%F a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^(3k-n).
%F a(n) = Sum_{k=0..floor(n/2)} binomial(k, n-2k)2^k(1/2)^(n-2k). (End)
%F From _Paul Barry_, Oct 25 2004: (Start)
%F G.f.: 1/((1+x)*(1-x-x^2)).
%F a(n) = Sum_{k=0..n} binomial(n-k-1, k). (End)
%F a(n) = |1 + (-1)^(n-1)*Fibonacci(n-1)|. - _Bridget Tenner_, Jun 04 2008
%F a(n) = A000045(n) + A033999(n). - _Michel Marcus_, Nov 14 2013
%F a(n) = Fibonacci(n+1) - a(n-1), with a(0) = 1. - _Franklin T. Adams-Watters_, Mar 26 2014
%F 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
%F 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
%F 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
%e 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.
%p with(combinat): f := n->fibonacci(n)+(-1)^n;
%t Table[Fibonacci[n]+(-1)^n,{n,0,50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2008 *)
%t CoefficientList[Series[1/(1-2x^2-x^3), {x, 0, 50}], x] (* _Vincenzo Librandi_, Jun 10 2013 *)
%t LinearRecurrence[{0,2,1}, {1,0,2}, 51] (* _Ray Chandler_, Sep 08 2015 *)
%o (Magma) [Fibonacci(n) + (-1)^n: n in [0..50]]; // _Vincenzo Librandi_, Apr 23 2011
%o (PARI) a(n)=fibonacci(n)+(-1)^n \\ _Charles R Greathouse IV_, Feb 03 2014
%o (Sage) [fibonacci(n)+(-1)^n for n in (0..50)] # _G. C. Greubel_, Jul 13 2019
%o (GAP) List([0..50], n-> Fibonacci(n) + (-1)^n) # _G. C. Greubel_, Jul 13 2019
%Y Cf. A007492, A066983, A078024, A119282, A014565, A099925, A098600, A374297.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_
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