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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Apparently, for n > 2, the same as A072337. - Ralf Stephan, Feb 01 2004

a(n) is the number of prime period-n 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)/((1-x)*(1-3*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_{d|n} 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)/((1-w)*(1-3*w+w^2)), w=0..y) = log((1-y)^2/(1-3*y+y^2)) for |y| < (3-sqrt(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_{d|gcd(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_{d|gcd(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(1-E(x^d)). For the sequence (e(n): n>=1) = (n: n>=1), we have E(x) = Sum_{n>=1} n*x^n = x/(1-x)^2, and thus A(x) = C(x) = -Sum_{d>=1} (mu(d)/d)*log(1-x/(1-x)^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(1-E(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((1-y)^2/(1-3*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

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Last modified October 19 09:57 EDT 2018. Contains 316349 sequences. (Running on oeis4.)