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A032170 "CHK" (necklace, identity, unlabeled) transform of 1, 2, 3, 4, ...

%I A032170 #90 Aug 18 2025 23:36:58
%S A032170 1,2,5,10,24,50,120,270,640,1500,3600,8610,20880,50700,124024,304290,
%T A032170 750120,1854400,4600200,11440548,28527320,71289000,178526880,
%U A032170 447910470,1125750120,2833885800,7144449920,18036373140,45591631800,115381697740,292329067800
%N A032170 "CHK" (necklace, identity, unlabeled) transform of 1, 2, 3, 4, ...
%C A032170 Apparently, for n > 2, the same as A072337. - _Ralf Stephan_, Feb 01 2004
%C A032170 a(n) is the number of prime period-n periodic orbits of Arnold's cat map. - _Bruce Boghosian_, Apr 26 2009
%C A032170 From _Petros Hadjicostas_, Nov 17 2017: (Start)
%C A032170 A first proof of the g.f., given below, can be obtained using the first of _Vladeta Jovovic_'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.
%C A032170 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.)
%C A032170 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)).
%C A032170 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.
%C A032170 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)
%C A032170 From _Petros Hadjicostas_, Jul 13 2020: (Start)
%C A032170 We elaborate further on _Michel Marcus_'s claim below. Consider his sequence (b(n): n >= 1) with b(1) = 3 and b(n) = a(n) for n >= 2.
%C A032170 Using the identity -Sum_{k >= 1} (mu(k)/k)*log(1 - x^k) = x for |x| < 1 and the g.f. of (a(n): n >= 1) below, we see that Sum_{n >= 1} b(n)*x^n = 3*x - a(1)*x + Sum_{n >= 1} a(n)*x^n = 2*x + Sum_{k >= 1} (mu(k)/k)*(2*log(1 - x^k) - log(1 - 3*x^k + x^(2*k))) = -Sum_{k >= 1} (mu(k)/k)*log(1 - 3*x^k + x^(2*k)).
%C A032170 Following Kam Cheong Au (2020), let d(w,N) be the dimension of the Q-span of weight w and level N of colored multiple zeta values (CMZV). Here Q are the rational numbers.
%C A032170 Deligne's bound says that d(w,N) <= D(w,N), where 1 + Sum_{w >= 1} D(w,N)*t^w = (1 - a*t + b*t^2)^(-1) when N >= 3, where a = phi(N)/2 + omega(N) and b = omega(N) - 1 (with omega(N) being the number of distinct primes of N).
%C A032170 For N = 6, a = phi(6)/2 + omega(6) = 2/2 + 2 = 3 and b = omega(6) - 1 = 1. It follows that D(w, N=6) = A001906(w+1) = Fibonacci(2*(w+1)).
%C A032170 For some reason, Kam Cheong Au (2020) assumes Deligne's bound is tight, i.e., d(w,N) = D(w,N). He sets Sum_{w >= 1} c(w,N)*t^w = log(1 + Sum_{w >= 1} d(w,N)*t^w) = log(1 + Sum_{w >= 1} D(w,N)*t^w) = -log(1 - a*t + b*t^2) for N >= 3.
%C A032170 For N = 6, we get that c(w, N=6) = A005248(w)/w.
%C A032170 He defines d*(w,N) = Sum_{k | w} (mu(k)/k)*c(w/k,N) to be the "number of primitive constants of weight w and level N". (Using the terminology of A113788, we may perhaps call d*(w,N) the number of irreducible colored multiple zeta values at weight w and level N.)
%C A032170 Using standard techniques of the theory of g.f.'s, we can prove that Sum_{w >= 1} d*(w,N)*t^w = Sum_{s >= 1} (mu(s)/s) Sum_{k >= 1} c(k,N)*(t^s)^k = -Sum_{s >= 1} (mu(s)/s)*log(1 - a*t^s + b*t^(2*s)).
%C A032170 For N = 6, we saw that a = 3 and b = 1, and hence d*(w, N=6) = b(w) for w >= 1 (as claimed by _Michel Marcus_ below). See Table 1 on p. 6 in Kam Cheong Au (2020). (End)
%H A032170 Michael De Vlieger, <a href="/A032170/b032170.txt">Table of n, a(n) for n = 1..2400</a>
%H A032170 Kam Cheong Au, <a href="https://arxiv.org/abs/2007.03957">Evaluation of one-dimensional polylogarithmic integral, with applications to infinite series</a>, arXiv:2007.03957 [math.NT], 2020. See line N = 6 in Table 1 (p. 6).
%H A032170 C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>.
%H A032170 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 A032170 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ArnoldsCatMap.html">Arnold's cat map</a>.
%H A032170 Wikipedia, <a href="http://en.wikipedia.org/wiki/Arnold&#39;s_cat_map">Arnold's cat map</a>.
%H A032170 <a href="/index/Lu#Lyndon">Index entries for sequences related to Lyndon words</a>
%F A032170 a(n) = (1/n)*Sum_{d | n} mu(n/d)*A004146(d). - _Vladeta Jovovic_, Feb 15 2003
%F A032170 Inverse EULER transform of Fibonacci(2*n). - _Vladeta Jovovic_, May 04 2006
%F A032170 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
%F A032170 It appears that the sequence b(1) = 3, b(n) = a(n) for n >= 2 is related to the rational sequence (c(w, N=6): w >= 1) = (A005248(w)/w: w >= 1) whose g.f. is log(1/(1 - a*t + b*t^2)), where a = phi(N)/2 + omega(N) and b = omega(N) - 1 when N = 6, where phi is A000010 and omega is A001221. See Kam Cheong Au (2020). - _Michel Marcus_, Jul 13 2020 [Edited by _Petros Hadjicostas_, Jul 13 2020]
%t A032170 Table[DivisorSum[n, MoebiusMu[n/#] (LucasL[2 #] - 2) &]/n, {n, 31}] (* _Michael De Vlieger_, Nov 18 2017 *)
%Y A032170 Cf. A000010, A001221, A001906, A004146, A032164, A032165, A032166, A032167, A032198, A005248, A108529, A113788.
%K A032170 nonn
%O A032170 1,2
%A A032170 _Christian G. Bower_