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Pisot sequence T(4,13), a(n) = floor(a(n-1)^2/a(n-2)).
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%I #25 Dec 14 2016 10:40:17

%S 4,13,42,135,433,1388,4449,14260,45706,146496,469546,1504979,4823727,

%T 15460908,49554976,158832563,509086778,1631714194,5229935889,

%U 16762880107,53728029453,172207945799,551957272549,1769121798104,5670351840955,18174492018967

%N Pisot sequence T(4,13), a(n) = floor(a(n-1)^2/a(n-2)).

%H Colin Barker, <a href="/A010919/b010919.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; with updates from 1996 and 1999.

%H D. G. Cantor, <a href="http://www.numdam.org/item?id=ASENS_1976_4_9_2_283_0">On families of Pisot E-sequences</a>, Ann. Sci. Ecole Nat. Sup. 9 (2) (1976) 283-308

%F Appears to satisfy the g.f. (4+x-x^2-x^4-x^36)/(1-3*x-x^2+x^3+x^5+x^37), where there is a common factor of 1+x that can be canceled, so the sequence appears to satisfy a linear recurrence of order 36. I believe that David Boyd has proved that the sequence does indeed satisfy this recurrence. - _N. J. A. Sloane_, Aug 11 2016

%t a[0] = 4; a[1] = 13; a[n_] := a[n] = Floor[a[n-1]^2/a[n-2]]; Array[a, 30, 0] (* _Jean-François Alcover_, Dec 14 2016 *)

%o (PARI) pisotT(nmax, a1, a2) = {

%o a=vector(nmax); a[1]=a1; a[2]=a2;

%o for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]));

%o a

%o }

%o pisotT(50, 4, 13) \\ _Colin Barker_, Jul 29 2016

%Y Cf. A022029, A010925.

%K nonn

%O 0,1

%A _Simon Plouffe_ and _N. J. A. Sloane_.