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A020708
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Pisot sequences E(4,9), P(4,9).
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2
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4, 9, 20, 44, 97, 214, 472, 1041, 2296, 5064, 11169, 24634, 54332, 119833, 264300, 582932, 1285697, 2835694, 6254320, 13794337, 30424368, 67103056, 148000449, 326425266, 719953588, 1587907625, 3502240516, 7724434620, 17036776865, 37575794246, 82876023112
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OFFSET
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0,1
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LINKS
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FORMULA
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a(n) = 2*a(n-1) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (4+x+2*x^2) / (1-2*x-x^3). - Colin Barker, Jun 05 2016
Theorem: E(4,9) satisfies a(n) = 2 a(n - 1) + a(n - 3) for n >= 3. Proved using the PtoRv program of Ekhad-Sloane-Zeilberger, and implies the above conjectures. - N. J. A. Sloane, Sep 09 2016
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MATHEMATICA
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RecurrenceTable[{a[0] == 4, a[1] == 9, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 30}] (* Bruno Berselli, Feb 05 2016 *)
LinearRecurrence[{2, 0, 1}, {4, 9, 20}, 40] (* Harvey P. Dale, Dec 19 2022 *)
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PROG
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(Magma) Exy:=[4, 9]; [n le 2 select Exy[n] else Floor(Self(n-1)^2/Self(n-2) + 1/2): n in [1..40]]; // Bruno Berselli, Feb 05 2016
(PARI) Vec((4+x+2*x^2) / (1-2*x-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020
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CROSSREFS
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See A008776 for definitions of Pisot sequences.
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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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