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A014739
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Expansion of (1+x^2)/(1-2*x+x^3).
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8
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Number of wedged n-spheres in the homotopy type of the Boolean complex of the affine Coxeter group A~ _n. - Bridget Eileen Tenner (bridget(AT)math.depaul.edu), Jun 04 2008
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 10 2009: (Start)
In an infinite set of sequences such than 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. (End)
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REFERENCES
| K. Ragnarsson and B. E. Tenner, Homotopy type of the Boolean complex of a Coxeter system
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LINKS
| Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1).
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FORMULA
| 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 (pbarry(AT)wit.ie), 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. [From Matthew Vandermast (ghodges14(AT)comcast.net), Nov 05 2009]
a(n)=2*a(n-1)-a(n-3). [From Vincenzo Librandi, Dec 31 2010]
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EXAMPLE
| 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
| 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
| CoefficientList[ Series[(1 + x^2)/(1 - 2*x + x^3), {x, 0, 35}], x] (from 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 [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 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] (* From Harvey P. Dale, June 26 2011 *)
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CROSSREFS
| Cf. A171516 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Dec 10 2009]
Sequence in context: A007979 A097701 A056870 * A039946 A130752 A059529
Adjacent sequences: A014736 A014737 A014738 * A014740 A014741 A014742
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Feb 25 2005
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