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A010908
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Pisot sequence E(4,21), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).
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1
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4, 21, 110, 576, 3016, 15792, 82688, 432960, 2267008, 11870208, 62153216, 325438464, 1704017920, 8922353664, 46718050304, 244618887168, 1280841121792, 6706571182080, 35116062605312, 183870090903552, 962756295000064, 5041057406386176, 26395319258316800
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OFFSET
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0,1
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COMMENTS
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For n >= 1, a(n-1) is the number of generalized compositions of n when there are i+3 different types of i, (i=1,2,...). - Milan Janjic, Sep 24 2010
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REFERENCES
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Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016.
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LINKS
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FORMULA
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a(n) = (((3-sqrt(5))^n*(-9+4*sqrt(5)) + (3+sqrt(5))^n*(9+4*sqrt(5))))/(2*sqrt(5)).
G.f.: (4-3*x) / (1-6*x+4*x^2).
(End)
Theorem: a(n) = 6*a(n-1) - 4*a(n-2) for n >= 2. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) This implies the above conjectures. - N. J. A. Sloane, Sep 09 2016
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MATHEMATICA
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RecurrenceTable[{a[1] == 4, a[2] == 21, a[n] == Floor[a[n-1]^2/a[n-2]+1/2]}, a, {n, 40}] (* Vincenzo Librandi, Aug 09 2016 *)
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PROG
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(PARI) a=vector(30); a[1]=4; a[2]=21; for(n=3, #a, a[n]=floor(a[n-1]^2/a[n-2]+1/2)); a \\ Colin Barker, Jun 04 2016
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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