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Pisot sequences E(3,7), P(3,7).
3

%I #72 Sep 08 2022 08:44:37

%S 3,7,16,37,86,200,465,1081,2513,5842,13581,31572,73396,170625,396655,

%T 922111,2143648,4983377,11584946,26931732,62608681,145547525,

%U 338356945,786584466,1828587033,4250949112,9882257736,22973462017,53406819691,124155792775

%N Pisot sequences E(3,7), P(3,7).

%H Vincenzo Librandi, <a href="/A010912/b010912.txt">Table of n, a(n) for n = 0..1000</a>

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa32/aa32110.pdf">Pisot sequences which satisfy no linear recurrences</a>, Acta Arith. 32 (1) (1977) 89-98.

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa34/aa3444.pdf">Some integer sequences related to the Pisot sequences</a>, Acta Arithmetica, 34 (1979), 295-305.

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa47/aa4712.pdf">On linear recurrence relations satisfied by Pisot sequences</a>, Acta Arithm. 47 (1) (1986) 13.

%H D. W. Boyd, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa48/aa4825.pdf">Pisot sequences which satisfy no linear recurrences. II</a>, Acta Arithm. 48 (1987) 191.

%H D. W. Boyd, <a href="https://www.researchgate.net/profile/David_Boyd7/publication/262181133_Linear_recurrence_relations_for_some_generalized_Pisot_sequences_-_annotated_with_corrections_and_additions/links/00b7d536d49781037f000000.pdf">Linear recurrence relations for some generalized Pisot sequences</a>, in Advances in Number Theory (Kingston ON, 1991), pp. 333-340, Oxford Univ. Press, New York, 1993.

%H S. B. Ekhad, N. J. A. Sloane, D. Zeilberger, <a href="http://arxiv.org/abs/1609.05570">Automated proofs (or disproofs) of linear recurrences satisfied by Pisot Sequences</a>, arXiv:1609.05570 [math.NT] (2016). [This is a different document from the one with the same title on Doron Zeilberger's web site]

%H Shalosh B. Ekhad, N. J. A. Sloane and Doron Zeilberger, <a href="http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/pisot.html">Automated Proof (or Disproof) of Linear Recurrences Satisfied by Pisot Sequences</a>, 2016; <a href="/A010912/a010912.pdf">Local copy</a> [pdf file only, no active links]

%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=895">Encyclopedia of Combinatorial Structures 895</a>.

%H C. Pisot, <a href="http://www.numdam.org/item?id=ASNSP_1938_2_7_3-4_205_0">La répartition modulo 1 et les nombres algébriques</a>, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1).

%F 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).

%F Empirical g.f.: (3-2*x+x^2)/(1-3*x+2*x^2-x^3). - _Colin Barker_, Feb 19 2012

%F 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

%F 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

%t a=1;b=1;c=1;Table[a+=b;b+=c;c+=a,{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Nov 17 2010 *)

%o (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

%o (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

%o (PARI) Vec((3-2*x+x^2)/(1-3*x+2*x^2-x^3) + O(x^30)) \\ _Jinyuan Wang_, Mar 10 2020

%Y See A008776 for definitions of Pisot sequences.

%K nonn,easy

%O 0,1

%A _Simon Plouffe_