A001869 Number of n-bead necklaces with 5 colors.
1, 5, 15, 45, 165, 629, 2635, 11165, 48915, 217045, 976887, 4438925, 20346485, 93900245, 435970995, 2034505661, 9536767665, 44878791365, 211927736135, 1003867701485, 4768372070757, 22706531350485, 108372083629275, 518301258916445
Offset: 0
References
- J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 162.
- N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).
Links
- T. D. Noe, Table of n, a(n) for n=0..200
- Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 21.
- E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
- INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 5
- Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
- J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
- Eric Weisstein's World of Mathematics, Necklace.
- Index entries for sequences related to necklaces
Programs
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Mathematica
CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-5*x^i]/i,{i,1,mx}],{x,0,mx}],x] (* Herbert Kociemba, Nov 01 2016 *) k=5; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *) -
PARI
a(n) = if (n, sumdiv(n, d, eulerphi(d)*5^(n/d))/n, 1); \\ Michel Marcus, Nov 01 2016
Formula
a(n) = (1/n)*Sum_{d|n} phi(d)*5^(n/d), n > 0.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 5*x^n)/n. - Herbert Kociemba, Nov 01 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 5^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 5^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
Comments