A014739 Expansion of (1+x^2)/(1-2*x+x^3).

1, 2, 5, 9, 16, 27, 45, 74, 121, 197, 320, 519, 841, 1362, 2205, 3569, 5776, 9347, 15125, 24474, 39601, 64077, 103680, 167759, 271441, 439202, 710645, 1149849, 1860496, 3010347, 4870845, 7881194, 12752041, 20633237, 33385280, 54018519, 87403801

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

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

Examples

			The Boolean complex of the affine Coxeter group \widetilde{A}_3 is homotopy equivalent to the wedge of 5 3-spheres.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system, Advances in Mathematics, Volume 222, Issue 2, 1 October 2009, Pages 409-430.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000032, A000045, A171516.

Programs

GAP

List([0..40], n-> Lucas(1,-1,n+2)[2] -2); # G. C. Greubel, Jul 22 2019

Magma

[Lucas(n+2)-2: n in [0..40]]; // G. C. Greubel, Jul 22 2019

Maple

with(combinat): seq(fibonacci(n)+fibonacci(n+2)-2, n=1..37); # Zerinvary Lajos, Jan 31 2008
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

Mathematica

CoefficientList[Series[(1+x^2)/(1-2*x+x^3), {x,0,40}], x] (* Robert G. Wilson v, Feb 25 2005 *)
a[0]=1; a[1]=2; a[2]=5; a[n_]:= a[n] = 2a[n-1]-a[n-3]; Array[a, 40, 0]
LinearRecurrence[{2,0,-1},{1,2,5},40] (* Harvey P. Dale, Jun 26 2011 *)
LucasL[Range[0,40]+2]-2 (* G. C. Greubel, Jul 22 2019 *)

PARI

Vec((1+x^2)/(1-2*x+x^3)+O(x^40)) \\ Charles R Greathouse IV, Sep 23 2012

PARI

vector(40, n, n--; f=fibonacci; f(n+3)+f(n+1)-2) \\ G. C. Greubel, Jul 22 2019

Sage

[lucas_number2(n+2,1,-1)-2 for n in (0..40)] # G. C. Greubel, Jul 22 2019

Formula

Partial sums of Lucas numbers A000032 less 1.

From Paul Barry, Sep 03 2003: (Start)

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

a(n) = ((3+sqrt(5))((1+sqrt(5))/2)^n+(3-sqrt(5))((1-sqrt(5))/2)^n)/2-2. (End)

From Zerinvary Lajos, Jan 31 2008: (Start)

a(n) = A001610(n+1)-1.

a(n) = F(n+1) + F(n+3) - 2 = A000071(n+1) + A000071(n+3), where F(n) is the n-th Fibonacci number. [corrected by R. J. Mathar, Mar 14 2011] (End)

a(n) = A000032(n+2) - 2. - Matthew Vandermast, Nov 05 2009

a(n) = 2*a(n-1) - a(n-3). - Vincenzo Librandi, Dec 31 2010

Extensions

More terms from Robert G. Wilson v, Feb 25 2005