A010912 Pisot sequences E(3,7), P(3,7).
3, 7, 16, 37, 86, 200, 465, 1081, 2513, 5842, 13581, 31572, 73396, 170625, 396655, 922111, 2143648, 4983377, 11584946, 26931732, 62608681, 145547525, 338356945, 786584466, 1828587033, 4250949112, 9882257736, 22973462017, 53406819691, 124155792775
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
- D. W. Boyd, Pisot sequences which satisfy no linear recurrences, Acta Arith. 32 (1) (1977) 89-98.
- D. W. Boyd, Some integer sequences related to the Pisot sequences, Acta Arithmetica, 34 (1979), 295-305.
- D. W. Boyd, On linear recurrence relations satisfied by Pisot sequences, Acta Arithm. 47 (1) (1986) 13.
- D. W. Boyd, Pisot sequences which satisfy no linear recurrences. II, Acta Arithm. 48 (1987) 191.
- D. W. Boyd, Linear recurrence relations for some generalized Pisot sequences, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, 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). [This is a different document from the one with the same title on Doron Zeilberger's web site]
- Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences, 2016; Local copy [pdf file only, no active links]
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 895.
- C. Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.
- Index entries for linear recurrences with constant coefficients, signature (3,-2,1).
Crossrefs
See A008776 for definitions of Pisot sequences.
Programs
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Magma
XY:=[3, 7]; [n le 2 select XY[n] else Ceiling(Self(n-1)^2/Self(n-2)-1/2): n in [1..32]]; // Klaus Brockhaus, Nov 17 2010
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Magma
a:=1; b:=1; c:=1; S:=[]; for n in [1..32] do a+:=b; b+:=c; c+:=a; Append(~S, c); end for; S; // Klaus Brockhaus, Nov 17 2010
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Mathematica
a=1;b=1;c=1;Table[a+=b;b+=c;c+=a,{n,50}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2010 *) -
PARI
Vec((3-2*x+x^2)/(1-3*x+2*x^2-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020
Formula
a(n) = 3*a(n-1) - 2*a(n-2) + a(n-3) (holds at least up to n = 1000 but is not known to hold in general).
Empirical g.f.: (3-2*x+x^2)/(1-3*x+2*x^2-x^3). - Colin Barker, Feb 19 2012
Since Pisot (1938) showed that E(3,k) always satisfies a linear recurrence, presumably it would not be difficult to prove that the above conjectures are correct. - N. J. A. Sloane, Jul 30 2016
Theorem: a(n) = 3 a(n - 1) - 2 a(n - 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