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A014739 Expansion of (1+x^2)/(1-2*x+x^3). 10
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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
LINKS
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.
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. - (End, corrected by R. J. Mathar, Mar 14 2011)
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
EXAMPLE
The Boolean complex of the affine Coxeter group \widetilde{A}_3 is homotopy equivalent to the wedge of 5 3-spheres.
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 *)
PROG
(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
(Magma) [Lucas(n+2)-2: n in [0..40]]; // 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
(GAP) List([0..40], n-> Lucas(1, -1, n+2)[2] -2); # G. C. Greubel, Jul 22 2019
CROSSREFS
Sequence in context: A211881 A056870 A326507 * A290464 A335768 A039946
KEYWORD
nonn,easy
AUTHOR
EXTENSIONS
More terms from Robert G. Wilson v, Feb 25 2005
STATUS
approved

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