%I #46 Jan 04 2024 03:21:16
%S 1,2,5,9,16,27,45,74,121,197,320,519,841,1362,2205,3569,5776,9347,
%T 15125,24474,39601,64077,103680,167759,271441,439202,710645,1149849,
%U 1860496,3010347,4870845,7881194,12752041,20633237,33385280,54018519,87403801
%N Expansion of (1+x^2)/(1-2*x+x^3).
%C Number of wedged n-spheres in the homotopy type of the Boolean complex of the affine Coxeter group A~ _n. - _Bridget Tenner_, Jun 04 2008
%C In an infinite set of sequences such that a(n) = a(n-1) + a(n-2) + k; with a(0) = 1, a(1) = 2, and in A014739, k = 2. Cf. A171516 for a(0) = 1, a(1) = 2, k = 3. - _Gary W. Adamson_, Dec 10 2009
%H G. C. Greubel, <a href="/A014739/b014739.txt">Table of n, a(n) for n = 0..1000</a>
%H K. Ragnarsson and B. E. Tenner, <a href="http://dx.doi.org/10.1016/j.aim.2009.05.007">Homotopy type of the Boolean complex of a Coxeter system</a>, Advances in Mathematics, Volume 222, Issue 2, 1 October 2009, Pages 409-430.
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).
%F Partial sums of Lucas numbers A000032 less 1.
%F From _Paul Barry_, Sep 03 2003: (Start)
%F G.f.: (1+x^2)/((1-x)*(1-x-x^2)).
%F a(n) = ((3+sqrt(5))((1+sqrt(5))/2)^n+(3-sqrt(5))((1-sqrt(5))/2)^n)/2-2. (End)
%F From _Zerinvary Lajos_, Jan 31 2008: (Start)
%F a(n) = A001610(n+1)-1.
%F a(n) = F(n+1) + F(n+3) - 2 = A000071(n+1) + A000071(n+3), where F(n) is the n-th Fibonacci number. - (End, corrected by _R. J. Mathar_, Mar 14 2011)
%F a(n) = A000032(n+2) - 2. - _Matthew Vandermast_, Nov 05 2009
%F a(n) = 2*a(n-1) - a(n-3). - _Vincenzo Librandi_, Dec 31 2010
%e The Boolean complex of the affine Coxeter group \widetilde{A}_3 is homotopy equivalent to the wedge of 5 3-spheres.
%p with(combinat): seq(fibonacci(n)+fibonacci(n+2)-2, n=1..37); # _Zerinvary Lajos_, Jan 31 2008
%p g:=(1+z^2)/(1-z-z^2): gser:=series(g, z=0, 43): seq(coeff(gser, z, n)-2, n=2..38); # _Zerinvary Lajos_, Jan 09 2009
%t CoefficientList[Series[(1+x^2)/(1-2*x+x^3), {x,0,40}], x] (* _Robert G. Wilson v_, Feb 25 2005 *)
%t a[0]=1; a[1]=2; a[2]=5; a[n_]:= a[n] = 2a[n-1]-a[n-3]; Array[a, 40, 0]
%t LinearRecurrence[{2,0,-1},{1,2,5},40] (* _Harvey P. Dale_, Jun 26 2011 *)
%t LucasL[Range[0,40]+2]-2 (* _G. C. Greubel_, Jul 22 2019 *)
%o (PARI) Vec((1+x^2)/(1-2*x+x^3)+O(x^40)) \\ _Charles R Greathouse IV_, Sep 23 2012
%o (PARI) vector(40, n, n--; f=fibonacci; f(n+3)+f(n+1)-2) \\ _G. C. Greubel_, Jul 22 2019
%o (Magma) [Lucas(n+2)-2: n in [0..40]]; // _G. C. Greubel_, Jul 22 2019
%o (Sage) [lucas_number2(n+2,1,-1)-2 for n in (0..40)] # _G. C. Greubel_, Jul 22 2019
%o (GAP) List([0..40], n-> Lucas(1,-1,n+2)[2] -2); # _G. C. Greubel_, Jul 22 2019
%Y Cf. A000032, A000045, A171516.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_
%E More terms from _Robert G. Wilson v_, Feb 25 2005