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 A010905 Pisot sequence E(4,15): a(n) = floor(a(n-1)^2/a(n-2)+1/2) for n>1, a(0)=4, a(1)=15. 2
 4, 15, 56, 209, 780, 2911, 10864, 40545, 151316, 564719, 2107560, 7865521, 29354524, 109552575, 408855776, 1525870529, 5694626340, 21252634831, 79315912984, 296011017105, 1104728155436, 4122901604639, 15386878263120, 57424611447841, 214311567528244 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 REFERENCES Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, Preprint, 2016. LINKS Colin Barker, Table of n, a(n) for n = 0..1000 D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305 D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, Advances in Number Theory ( Kingston ON, 1991) 333-340, Oxford Sci. Publ., Oxford Univ. Press, New York, 1993. Tanya Khovanova, Recursive Sequences Index entries for linear recurrences with constant coefficients, signature (4,-1). FORMULA a(n) = 4*a(n-1) - a(n-2) for n>=2. (Proved using the PtoRv program of Ekhad-Sloane-Zeilberger.) - N. J. A. Sloane, Sep 09 2016 This was conjectured by Colin Barker, Apr 16 2012, and implies the G.f.: (4-x)/(1-4*x+x^2) and the formula a(n) = ((1+sqrt(3))^(2*n+4)-(1-sqrt(3))^(2*n+4))/(2^(n+3)*sqrt(3)). Partial sums of A079935. - Erin Pearse, Dec 13 2018 MATHEMATICA a = 4; a = 15; a[n_] := a[n] = Floor[a[n - 1]^2/a[n - 2] + 1/2]; Table[a[n], {n, 0, 24}] (* Michael De Vlieger, Jul 27 2016 *) PROG (MAGMA) /* By definition: */ [n le 2 select 11*n-7 else Floor(Self(n-1)^2/Self(n-2)+1/2): n in [1..22]]; // Bruno Berselli, Apr 16 2012 (PARI) pisotE(nmax, a1, a2) = {   a=vector(nmax); a=a1; a=a2;   for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]+1/2));   a } pisotE(50, 4, 15) \\ Colin Barker, Jul 27 2016 (Sage) @cached_function def A010905(n):     if n==0: return 4     elif n==1: return 15     else: return 4*A010905(n-1) - A010905(n-2) [A010905(n) for n in range(30)] # G. C. Greubel, Dec 13 2018 CROSSREFS Cf. A010925, A001353, A079935, A195503. Sequence in context: A106707 A125905 A195503 * A026030 A047038 A158500 Adjacent sequences:  A010902 A010903 A010904 * A010906 A010907 A010908 KEYWORD nonn AUTHOR EXTENSIONS Edited by N. J. A. Sloane, Jul 26 2016 and Sep 09 2016 STATUS approved

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Last modified October 25 13:18 EDT 2020. Contains 338012 sequences. (Running on oeis4.)