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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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LINKS
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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. (End)
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)
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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, 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
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MATHEMATICA
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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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