%I #45 May 17 2018 03:09:52

%S 5,21,88,368,1538,6427,26857,112229,468978,1959746,8189306,34221135,

%T 143001871,597570335,2497102330,10434788478,43604464772,182212543365,

%U 761422279419,3181800093939,13295975323332,55560674643076,232174661258332,970201922073653,4054239874815929,16941690784755705,70795240417122019

%N Pisot sequence T(5,21), a(n) = floor( a(n-1)^2/a(n-2) ).

%C Comments from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004, with updates from _N. J. A. Sloane_, Aug 05 2016: (Start)

%C Different from A019992. The two sequences differ from n=26 on (A010925(26) = 70795240417122019 != 70795240417122020 = A019992(26)).

%C From Boyd's paper "Linear recurrence relations for some generalized Pisot sequences", T(5,21) satisfies the rational generating function F(x)/(1+x-x*F(x)), with F(x) = 5 + x - x^2 - x^4 - x^26 - x^2048, a 2049th-order recurrence; and not the A019992 generating function: F(x)/(1+x-x*F(x)), with F(x) = 5 + x - x^2 - x^4, which gives the 5th-order recurrence for A019992.

%C The g.f. F(x)/(1+x-x*F(x)) with F(x) = 5 + x - x^2 - x^4 - x^26 - x^2048 is not in lowest terms, however, and a factor of 1+x can be canceled. The lowest-order recurrence satisfied by this sequence has order 2048.

%C This and other examples show that it is essential to reject conjectured generating functions for Pisot sequences until a proof or reference is provided. (End)

%H Colin Barker, <a href="/A010925/b010925.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) 13-27; 54 (1990), 255-256.

%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) 191-195.

%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

%H M. J. DeLeon, <a href="http://dx.doi.org/10.1515/crll.1971.249.20">Pisot Sequences</a>, J. Reine Angew. Mathem. 249 (1971) 20-30

%H Charles Pisot, <a href="http://archive.numdam.org/article/THESE_1938__203__1_0.pdf">La répartition modulo un et les nombres algébriques</a>, Thesis (1938).

%H Charles 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.

%F G.f.: F(x)/(1+x-x*F(x)), with F(x) = 5 + x - x^2 - x^4 - x^26 - x^2048 (D. W. Boyd). - Pab Ter (pabrlos(AT)yahoo.com), May 23 2004

%t nxt[{a_,b_}]:={b,Floor[b^2/a]}; NestList[nxt,{5,21},30][[All,1]] (* _Harvey P. Dale_, May 15 2017 *)

%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, 5, 21) \\ _Colin Barker_, Jul 27 2016

%K nonn

%O 0,1

%A _Simon Plouffe_

%E More terms from Pab Ter (pabrlos(AT)yahoo.com), May 23 2004