A010908 Pisot sequence E(4,21), a(n) = floor(a(n-1)^2/a(n-2) + 1/2).

4, 21, 110, 576, 3016, 15792, 82688, 432960, 2267008, 11870208, 62153216, 325438464, 1704017920, 8922353664, 46718050304, 244618887168, 1280841121792, 6706571182080, 35116062605312, 183870090903552, 962756295000064, 5041057406386176, 26395319258316800

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, and 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, and D. Zeilberger, Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences, arXiv:1609.05570 [math.NT] (2016)

Index entries for linear recurrences with constant coefficients, signature (6,-4).

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

a(n) = 2^(n-1)*A000045(2*n+6). - Taras Goy, Jan 05 2025

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

Simon Plouffe

Status

approved