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A125905
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a(0) = 1, a(1) = -4, a(n) = -4*a(n-1) - a(n-2) for n > 1.
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10
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1, -4, 15, -56, 209, -780, 2911, -10864, 40545, -151316, 564719, -2107560, 7865521, -29354524, 109552575, -408855776, 1525870529, -5694626340, 21252634831, -79315912984, 296011017105, -1104728155436, 4122901604639, -15386878263120, 57424611447841
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OFFSET
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0,2
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COMMENTS
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Pisano period lengths: 1, 2, 3, 4, 6, 6, 8, 4, 9, 6, 5, 12, 12, 8, 6, 8, 9, 18, 10, 12, ... - R. J. Mathar, Aug 10 2012
In engineering literature, these numbers are known as Clapeyron numbers, or Clapeyron's numbers, or Clapeyronian numbers, on account of their appearance in Benoît Clapeyron's influential study (1857) of the bending forces imposed upon multiple supports of a horizontal beam. - John Blythe Dobson, Mar 12 2014
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REFERENCES
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Harold J. Ahlberg, Edwin N. Nilson and Joseph L. Walsh, The Theory of Splines and Their Applications, Academic Press, 1967, pp. 35-46.
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LINKS
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FORMULA
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G.f.: 1/(1 + 4*x + x^2).
a(n) = (-2)^n*((1 + sqrt(3)/2)^(n + 1) - (1 - sqrt(3)/2)^(n + 1))/sqrt(3).
E.g.f.: exp(-2*x)*(3*cosh(sqrt(3)*x) - 2*sqrt(3)*sinh(sqrt(3)*x))/3. (End)
a(n) = (-2)^n*Product_{k=1..n}(2 + cos(k*Pi/(n+1))). - Peter Luschny, Nov 28 2019
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MATHEMATICA
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CoefficientList[Series[1/(1+4*x+x^2), {x, 0, 50}], x] (* Vincenzo Librandi, Jun 28 2012 *)
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PROG
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(Magma) I:=[1, -4]; [n le 2 select I[n] else -4*Self(n-1)-Self(n-2): n in [1..30]]; // Vincenzo Librandi, Jun 28 2012
(PARI) x='x+O('x^30); Vec(1/(1+4*x+x^2)) \\ G. C. Greubel, Feb 05 2018
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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EXTENSIONS
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Typo in a(22) corrected by Neven Juric, Dec 20 2010
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STATUS
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approved
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