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A060167 Number of orbits of length n under the map whose periodic points are counted by A001642.

%I A060167 #11 Jan 05 2025 19:51:36
%S A060167 1,1,1,2,4,5,9,13,23,36,63,101,175,290,497,840,1445,2460,4247,7293,
%T A060167 12619,21805,37856,65695,114401,199280,347944,607959,1064130,1864083,
%U A060167 3269948,5740840,10090148,17748870,31250297,55063603,97102485,171355485,302605780,534729160,945513850
%N A060167 Number of orbits of length n under the map whose periodic points are counted by A001642.
%C A060167 The sequence A001642 seems to record the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.
%H A060167 Y. Puri and T. Ward, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/WARD/short.html">Arithmetic and growth of periodic orbits</a>, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
%H A060167 Yash Puri and Thomas Ward, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/39-5/puri.pdf">A dynamical property unique to the Lucas sequence</a>, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
%F A060167 a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001642(n/d).
%e A060167 u(7) = 9 since a map whose periodic points are counted by A001642 would have 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.
%o A060167 (PARI) a001642(n) = if(n<0, 0, polcoeff(x*(1+2*x+4*x^3+5*x^4)/(1-x-x^2-x^4-x^5)+x*O(x^n), n));
%o A060167 a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001642(n/d)); \\ _Michel Marcus_, Sep 11 2017
%Y A060167 Cf. A001642, A060164, A060165, A060166, A060168, A060169, A060170, A060171, A060171.
%K A060167 easy,nonn
%O A060167 1,4
%A A060167 _Thomas Ward_, Mar 13 2001
%E A060167 More terms from _Michel Marcus_, Sep 11 2017