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 A020709 Pisot sequence E(4,10). 1
 4, 10, 25, 63, 159, 401, 1011, 2549, 6427, 16205, 40859, 103021, 259755, 654941, 1651355, 4163693, 10498251, 26470077, 66741115, 168279693, 424297003, 1069813853, 2697406939, 6801187117, 17148375179, 43237565181, 109018319419, 274876578253, 693068226283 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 LINKS Colin Barker, Table of n, a(n) for n = 0..1000 Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT], 2016. Index entries for linear recurrences with constant coefficients, signature (3,-2,2). FORMULA a(n) = 3*a(n-1) - 2*a(n-2) + 2*a(n-3) (holds at least up to n = 1000 but is not known to hold in general). Empirical g.f.: (4-2*x+3*x^2) / (1-3*x+2*x^2-2*x^3). - Colin Barker, Jun 05 2016 Theorem: a(n) = 3 a(n - 1) - 2 a(n - 2) + 2 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 MATHEMATICA RecurrenceTable[{a[0] == 4, a[1] == 10, a[n] == Floor[a[n - 1]^2/a[n - 2] + 1/2]}, a, {n, 0, 40}] (* Bruno Berselli, Feb 05 2016 *) LinearRecurrence[{3, -2, 2}, {4, 10, 25}, 30] (* Harvey P. Dale, Jan 29 2021 *) PROG (MAGMA) Exy:=[4, 10]; [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((4-2*x+3*x^2)/(1-3*x+2*x^2-2*x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020 CROSSREFS See A008776 for definitions of Pisot sequences. Sequence in context: A033539 A020748 A021004 * A020734 A225301 A298412 Adjacent sequences:  A020706 A020707 A020708 * A020710 A020711 A020712 KEYWORD nonn AUTHOR STATUS approved

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Last modified April 21 14:29 EDT 2021. Contains 343154 sequences. (Running on oeis4.)