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 A010908 Pisot sequence E(4,21), a(n) = floor(a(n-1)^2/a(n-2) + 1/2). 1
 4, 21, 110, 576, 3016, 15792, 82688, 432960, 2267008, 11870208, 62153216, 325438464, 1704017920, 8922353664, 46718050304, 244618887168, 1280841121792, 6706571182080, 35116062605312, 183870090903552, 962756295000064, 5041057406386176, 26395319258316800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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 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 Daniel Birmajer, Juan B. Gil, Michael D. Weiner, (an + b)-color compositions, arXiv:1707.07798 [math.CO], 2017. 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. S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016) FORMULA Conjectures from Colin Barker, Jun 04 2016: (Start) 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 MATHEMATICA 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 *) PROG (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 CROSSREFS Sequence in context: A183367 A100237 A117381 * A136786 A301972 A026335 Adjacent sequences:  A010905 A010906 A010907 * A010909 A010910 A010911 KEYWORD nonn AUTHOR STATUS approved

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Last modified March 24 01:20 EDT 2019. Contains 321444 sequences. (Running on oeis4.)