A052823 A simple grammar: cycles of pairs of sequences.
0, 0, 1, 2, 4, 6, 12, 18, 34, 58, 106, 186, 350, 630, 1180, 2190, 4114, 7710, 14600, 27594, 52486, 99878, 190744, 364722, 699250, 1342182, 2581426, 4971066, 9587578, 18512790, 35792566, 69273666, 134219794, 260301174, 505294126, 981706830, 1908881898
Offset: 0
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 788
- Terry R. Payne, Luke Riley, Katie Atkinson, and Paul Dunne, Using Two Colour Necklaces to Fairly Allocate Coalition Value Calculations, 23rd IEEE/WIC Int'l Conf. Web Intel. Intel. Agent Tech. (WI-IAT 2024). See p. 7.
- Shingo Saito, Tatsushi Tanaka, and Noriko Wakabayashi, Combinatorial Remarks on the Cyclic Sum Formula for Multiple Zeta Values, J. Int. Seq. 14 (2011) # 11.2.4, Table 2.
Programs
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Maple
spec := [S,{B=Sequence(Z,1 <= card),C=Prod(B,B),S= Cycle(C)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20); -
Mathematica
k=2; Prepend[Table[k!DivisorSum[n,EulerPhi[#]StirlingS2[n/#,k]&]/n,{n,1,30}],0] (* Robert A. Russell, Sep 26 2018 *)
Formula
G.f.: Sum_{j>=1} phi(j)/j*log(-(x^j-1)^2/(2*x^j-1)).
a(n) = (k!/n) Sum_{d|n} phi(d) S2(n/d,k), where k=2 is the number of colors and S2 is the Stirling subset number A008277. - Robert A. Russell, Sep 26 2018
a(n) ~ 2^n / n. - Vaclav Kotesovec, Sep 25 2019
Extensions
More terms from Alois P. Heinz, Jan 25 2015
Comments