A010925 Pisot sequence T(5,21), a(n) = floor( a(n-1)^2/a(n-2) ).

5, 21, 88, 368, 1538, 6427, 26857, 112229, 468978, 1959746, 8189306, 34221135, 143001871, 597570335, 2497102330, 10434788478, 43604464772, 182212543365, 761422279419, 3181800093939, 13295975323332, 55560674643076, 232174661258332, 970201922073653, 4054239874815929, 16941690784755705, 70795240417122019

Offset: 0

Author

Simon Plouffe

Keywords

• nonn

Comments

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

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

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.

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.

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)

Links

• Colin Barker, 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) 13-27; 54 (1990), 255-256.
• D. W. Boyd, Pisot sequences which satisfy no linear recurrences. II, Acta Arithm. 48 (1987) 191-195.
• 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; with updates from 1996 and 1999.
• D. G. Cantor, On families of Pisot E-sequences, Ann. Sci. Ecole Nat. Sup. 9 (2) (1976) 283-308
• M. J. DeLeon, Pisot Sequences, J. Reine Angew. Mathem. 249 (1971) 20-30
• Charles Pisot, La répartition modulo un et les nombres algébriques, Thesis (1938).
• Charles Pisot, La répartition modulo 1 et les nombres algébriques, Ann. Scuola Norm. Sup. Pisa, 7 (1938), 205-248.

Programs

Mathematica

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

PARI

pisotT(nmax, a1, a2) = {
  a=vector(nmax); a[1]=a1; a[2]=a2;
  for(n=3, nmax, a[n] = floor(a[n-1]^2/a[n-2]));
  a
}
pisotT(50, 5, 21) \\ Colin Barker, Jul 27 2016

Formula

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

Extensions

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