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Number of points of period n under the dual of the map x->2x on Z[1/6].
2

%I #7 Feb 19 2021 20:10:00

%S 1,1,7,5,31,7,127,85,511,341,2047,455,8191,5461,32767,21845,131071,

%T 9709,524287,349525,2097151,1398101,8388607,1864135,33554431,22369621,

%U 134217727,89478485,536870911,119304647

%N Number of points of period n under the dual of the map x->2x on Z[1/6].

%C This sequence counts the periodic points in the simplest nontrivial S-integer dynamical system. These dynamical systems arise naturally in arithmetic and are built by making an isometric extension of a familiar hyperbolic system. The extension destroys some of the periodic points, in this case reducing the original number 2^n-1 by factoring out any 3's. An interesting feature is that the logarithmic growth rate is still log 2.

%C A059990[n+7] times some power of 3 seems to me the greatest common Denominator of A035522[4n+16+1],A035522[4n+16+2],A035522[4n+16+3] and A035522[4n+16+4] for n>1 [From _Dylan Hamilton_, Aug 04 2010]

%D V. Chothi, G. Everest, T. Ward. S-integer dynamical systems: periodic points. J. Reine Angew. Math., 489 (1997), 99-132.

%D T. Ward. Almost all S-integer dynamical systems have many periodic points. Erg. Th. Dynam. Sys. 18 (1998), 471-486.

%H Y. Puri and T. Ward, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.

%F a(n)=(2^n-1)x|2^n-1|_3

%e a(6)=7 because 2^6-1 = 3^2x7, so |2^6-1|_3=3^(-2).

%Y Cf. A000225, A001945, A059991.

%K easy,nonn

%O 1,3

%A _Thomas Ward_, Mar 08 2001