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A010912 Pisot sequences E(3,7), P(3,7). 3
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

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).

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

MATHEMATICA

a=1; b=1; c=1; Table[a+=b; b+=c; c+=a, {n, 50}] (* Vladimir Joseph Stephan Orlovsky, Nov 17 2010 *)

PROG

(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

(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

(PARI) Vec((3-2*x+x^2)/(1-3*x+2*x^2-x^3) + O(x^30)) \\ Jinyuan Wang, Mar 10 2020

CROSSREFS

See A008776 for definitions of Pisot sequences.

Sequence in context: A188626 A123392 A095263 * A192665 A052967 A297498

Adjacent sequences:  A010909 A010910 A010911 * A010913 A010914 A010915

KEYWORD

nonn,easy

AUTHOR

Simon Plouffe

STATUS

approved

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Last modified July 24 16:24 EDT 2021. Contains 346273 sequences. (Running on oeis4.)