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A014739
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Expansion of (1+x^2)/(1-2*x+x^3).
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10
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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
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OFFSET
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0,2
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COMMENTS
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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
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REFERENCES
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K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system
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LINKS
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Table of n, a(n) for n=0..36.
Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1).
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FORMULA
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Partial sums of Lucas numbers A000032 less 1.
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. - Paul Barry, Sep 03 2003
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. - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 31 2008, 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]
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EXAMPLE
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The Boolean complex of the affine Coxeter group \widetilde{A}_3 is homotopy equivalent to the wedge of 5 3-spheres.
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MAPLE
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with(combinat): seq(fibonacci(n)+fibonacci(n+2)-2, n=1..37); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jan 09 2009]
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MATHEMATICA
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CoefficientList[ Series[(1 + x^2)/(1 - 2*x + x^3), {x, 0, 35}], x] (* Robert G. Wilson v, Feb 25 2005)
a=0; lst={}; s=0; Do[a=s-(a-1); AppendTo[lst, Abs[a]]; s+=a-2, {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 27 2009 *)
a[0] = 1; a[1] = 2; a[2] = 5; a[n_] := a[n] = 2 a[n - 1] - a[n - 3]; Array[a, 37, 0]
LinearRecurrence[{2, 0, -1}, {1, 2, 5}, 50] (* Harvey P. Dale, Jun 26 2011 *)
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PROG
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(PARI) Vec((1+x^2)/(1-2*x+x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012
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CROSSREFS
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Cf. A171516.
Sequence in context: A097701 A211881 A056870 * A039946 A130752 A059529
Adjacent sequences: A014736 A014737 A014738 * A014740 A014741 A014742
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KEYWORD
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nonn,easy
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Robert G. Wilson v, Feb 25 2005
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STATUS
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approved
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