A060166 Number of orbits of length n under the map whose periodic points are counted by A001641.
1, 1, 1, 2, 3, 4, 7, 10, 17, 26, 44, 68, 115, 184, 306, 500, 835, 1374, 2301, 3822, 6409, 10718, 18028, 30280, 51077, 86130, 145641, 246370, 417600, 708246, 1203069, 2045010, 3480408, 5927660, 10105819, 17241140, 29439580, 50302162, 86012630, 147166248, 251963055, 431633348
Offset: 1
Examples
a(7) = 7 since a map whose periodic points are counted by A001641 would have 1 fixed point and 50 points of period 7, hence 7 orbits of length 7.
Links
- Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
- Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Programs
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PARI
a001641(n)=if(n<0, 0, polcoeff(x*(1+2*x+4*x^3)/(1-x-x^2-x^4)+x*O(x^n), n)) a(n) = sumdiv(n, d, moebius(d)*a001641(n/d))/n; \\ Michel Marcus, Sep 10 2017
Formula
a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001641(n/d).
Extensions
More terms from Michel Marcus, Sep 10 2017
Comments