

A032170


"CHK" (necklace, identity, unlabeled) transform of 1,2,3,4,...


5



1, 2, 5, 10, 24, 50, 120, 270, 640, 1500, 3600, 8610, 20880, 50700, 124024, 304290, 750120, 1854400, 4600200, 11440548, 28527320, 71289000, 178526880, 447910470, 1125750120, 2833885800, 7144449920, 18036373140, 45591631800, 115381697740, 292329067800
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OFFSET

1,2


COMMENTS

Apparently, for n > 2, the same as A072337.  Ralf Stephan, Feb 01 2004
a(n) is the number of prime periodn periodic orbits of Arnold's cat map.  Bruce Boghosian, Apr 26 2009
From Petros Hadjicostas, Nov 17 2017: (Start)
A first proof of the g.f., given below, can be obtained using the first of V. Jovovich's formulae. If b(n) = A004146(n), then B(x) = Sum_{n>=1} b(n)*x^n = x*(1+x)/((1x)*(13*x+x^2)) (see the documentation for sequence A004146). From Jovovich's first formula, A(x) = Sum_{n>=1} a(n)*x^n = Sum_{n>=1} (1/n)*Sum_{dn} mu(d)*b(n/d)*x^n. Letting m=n/d, we get A(x) = Sum_{d>=1} (mu(d)/d)*Sum_{m>=1} b(m)*(x^d)^m/m = Sum_{d>=1} (mu(d)/d)*f(x^d), where f(y) = Sum_{m>=1} b(m)*y^m/m = int(B(w)/w, w=0..y) = int((1+w)/((1w)*(13*w+w^2)), w=0..y) = log((1y)^2/(13*y+y^2)) for y < (3sqrt(5))/2.
A second proof of the g.f. can be obtained using C. G. Bower's definition of the CHK transform of a sequence (e(n): n>=1) with g.f. E(x) (see the links below). If (c_k(n): n>=1) = CHK_k(e(n): n>=1), then (c_k(n): n>=1) = (1/k)*(MOEBIUS*AIK)_k (e_n: n>=1) = (1/k)*Sum_{dgcd(n,k)} mu(d)*AIK_{k/d}(e(n/d): n multiple of d), where the * between MOEBIUS and AIK denotes Dirichlet convolution and (d_k(n): n>=1) = AIK_k(e(n): n>=1) has g.f. E(x)^k. (There is a typo in the given definition of CHK in the link.)
If C(x) is the g.f. of CHK(e(n): n>=1) = Sum_{k=1..n} CHK_k(e(n): n>=1), then C(x) = Sum_{n>=1} Sum_{k=1..n} c_k(n)*x^n = Sum_{k>=1} (1/k) Sum_{n>=k} Sum_{dgcd(n,k)} mu(d)*d_{k/d}(n/d)*x^n. Letting m=n/d and s=k/d and using the fact that E(0)=0, we get C(x) = Sum_{d>=1} (mu(d)/d)*Sum_{s>=1} (1/s)*Sum_{m>=s} d_s(m)*(x^d)^m = Sum_{d>=1} (mu(d)/d)*Sum_{s>=1} E(x^d)^s. Thus, C(x) = Sum_{d>=1} (mu(d)/d)*log(1E(x^d)). For the sequence (e(n): n>=1) = (n: n>=1), we have E(x) = Sum_{n>=1} n*x^n = x/(1x)^2, and thus A(x) = C(x) = Sum_{d>=1} (mu(d)/d)*log(1x/(1x)^2), from which we can easily get the g.f. given in the formula section.
Apparently, for this sequence and for sequences A032165, A032166, A032167, the author assumes that C(0) = 0 (i.e., he assumes the CHK transform has no constant term), while for sequences A032164, A108529, and possibly others, he assumes that the CHK transform starts with the constant term 1 (i.e., he assumes C(x) = 1  Sum_{d>=1} (mu(d)/d)*log(1E(x^d))).
(End)


LINKS

Michael De Vlieger, Table of n, a(n) for n = 1..2400
C. G. Bower, Transforms (2)
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Eric Weisstein's World of Mathematics, Arnold's cat map
Wikipedia, Arnold's cat map [From Bruce Boghosian, Apr 26 2009]
Index entries for sequences related to Lyndon words


FORMULA

a(n) = 1/n*Sum_{d divides n} mu(n/d)*A004146(d).  Vladeta Jovovic, Feb 15 2003
Inverse EULER transform of Fibonacci(2*n).  Vladeta Jovovic, May 04 2006
G.f.: Sum_{n>=1} (mu(n)/n)*f(x^n), where f(y) = log((1y)^2/(13*y+y^2)).  Petros Hadjicostas, Nov 17 2017


MATHEMATICA

Table[DivisorSum[n, MoebiusMu[n/#] (LucasL[2 #]  2) &]/n, {n, 31}] (* Michael De Vlieger, Nov 18 2017 *)


CROSSREFS

Cf. A032198.
Sequence in context: A026754 A291247 A316697 * A084081 A106376 A151514
Adjacent sequences: A032167 A032168 A032169 * A032171 A032172 A032173


KEYWORD

nonn


AUTHOR

Christian G. Bower


STATUS

approved



