|
|
A059990
|
|
Number of points of period n under the dual of the map x->2x on Z[1/6].
|
|
2
|
|
|
1, 1, 7, 5, 31, 7, 127, 85, 511, 341, 2047, 455, 8191, 5461, 32767, 21845, 131071, 9709, 524287, 349525, 2097151, 1398101, 8388607, 1864135, 33554431, 22369621, 134217727, 89478485, 536870911, 119304647
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
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.
|
|
REFERENCES
|
V. Chothi, G. Everest, T. Ward. S-integer dynamical systems: periodic points. J. Reine Angew. Math., 489 (1997), 99-132.
T. Ward. Almost all S-integer dynamical systems have many periodic points. Erg. Th. Dynam. Sys. 18 (1998), 471-486.
|
|
LINKS
|
|
|
FORMULA
|
a(n)=(2^n-1)x|2^n-1|_3
|
|
EXAMPLE
|
a(6)=7 because 2^6-1 = 3^2x7, so |2^6-1|_3=3^(-2).
|
|
CROSSREFS
|
|
|
KEYWORD
|
easy,nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|