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A158563
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a(n) = 32*n^2 - 1.
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4
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31, 127, 287, 511, 799, 1151, 1567, 2047, 2591, 3199, 3871, 4607, 5407, 6271, 7199, 8191, 9247, 10367, 11551, 12799, 14111, 15487, 16927, 18431, 19999, 21631, 23327, 25087, 26911, 28799, 30751, 32767, 34847, 36991, 39199, 41471, 43807, 46207, 48671, 51199, 53791
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OFFSET
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1,1
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COMMENTS
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The identity (32*n^2-1)^2 - (256*n^2-16)*(2*n)^2 = 1 can be written as a(n)^2 - A158562(n)*A005843(n)^2 = 1. [comment rewritten by R. J. Mathar, Oct 16 2009]
Sequence found by reading the line from 31, in the direction 31, 127, ..., in the rectangular spiral whose vertices are the generalized 18-gonal numbers A274979.
The spiral begins as follows:
46_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _18
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| 0 |
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| 1 15
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51
(End)
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LINKS
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Vincenzo Librandi, X^2-AY^2=1, Math Forum, 2007. [Wayback Machine link]
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FORMULA
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G.f.: x*(-31-34*x+x^2)/(x-1)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = (1 - cot(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)))/2.
Sum_{n>=1} (-1)^(n+1)/a(n) = (cosec(Pi/(4*sqrt(2)))*Pi/(4*sqrt(2)) - 1)/2. (End)
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MATHEMATICA
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CoefficientList[Series[(- 31 - 34 x + x^2) / (x - 1)^3, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)
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PROG
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CROSSREFS
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Cf. A274979 (generalized 18-gonal numbers).
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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