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)
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
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
KEYWORD
nonn
AUTHOR
STATUS
approved