0,2

Comment from Pab Ter: "The number of equivalence classes of sequences of period p, taking values in a set with b elements, is given by:

N(p) = 1/(p*phi(p)) sum_{0<=t<=p-1} sum_{1<=k<=p-1 & gcd(p,k)=1} b^C(k,t) where C(k,t), the number of disjoint cycles of the permutations considered, is C(k,t) = sum_{0<=u<=p-1} 1/M(k,p/gcd(p,u(k-1)+t))

If gcd(k,L)=1, M(k,L) denotes the least positive integer M such that 1+k+..+k^(M-1) == 0 (mod L). Also if gcd(k,L)=1 and Ek(L) denotes the exponent of k mod L: M(k,L)=L*Ek(L)/gcd(L,1+k+..+k^(Ek(L)-1))."

Number of two-colored necklaces of length n, where similar necklaces are counted only once. Two necklaces of length n, given by color functions c and d from {0, ..., n-1} to N (set of natural numbers) are considered similar iff there is a factor f, 0 < f < n, satisfying GCD (f,n) = 1, such that, for all k from {0, ..., n-1}, d(f * k mod n) = c(k). I.e. the bead at position k is moved to f * k mod n. In other words: the sequence counts the orbits of the action of the multiplicative group {f | 0 < f < n, GCD (f,n) = 1} on the set of two-colored necklaces where f maps c to d with the formula above. - Matthias Engelhardt (Matthias.R.Engelhardt(AT)web.de)

Counts the same necklaces as A000029 but some of the necklaces viewed as distinct in A000029 are now viewed as equal. In particular, this implies that a(n) <= A000029(n) for every n.

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. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

M. Engelhardt, The N queens problem

Index entries for sequences related to necklaces

Reference gives formula.

with(numtheory): M:=proc(k, L) local e, s: s:=1: for e from 1 do if(s mod L = 0) then RETURN(e) else s:=s+k^e fi od: end; C:=proc(k, t, p) local u: RETURN(add(M(k, p/igcd(p, u*(k-1)+t))^(-1), u=0..p-1)) :end; N:=proc(p) options remember: local s, t, k: if(p=1) then RETURN(2) fi: s:=0: for t from 0 to p-1 do for k from 1 to p-1 do if igcd(p, k)=1 then s:=s+2^C(k, t, p) fi od od: RETURN(s/(p*phi(p))):end; seq(N(p), p=1..51); # first M expression with(numtheory): E:=proc(k, L) if(L=1) then RETURN(1) else RETURN(order(k, L)) fi end; M:=proc(k, L) local s, EkL: EkL:=E(k, L): if(k>1) then s:=(k^EkL-1)/(k-1): RETURN(L*EkL/igcd(L, s)) else RETURN(L*EkL/igcd(L, EkL)) fi end; C:=proc(k, t, p) local u: RETURN(add(M(k, p/igcd(p, u*(k-1)+t))^(-1), u=0..p-1)) :end; N:=proc(p) options remember: local s, t, k: if(p=1) then RETURN(2) fi: s:=0: for t from 0 to p-1 do for k from 1 to p-1 do if igcd(p, k)=1 then s:=s+2^C(k, t, p) fi od od: RETURN(s/(p*phi(p))):end; seq(N(p), p=1..51); # second M expression (Pab Ter)

Cf. A002730.

Sequence in context: A123131 A000793 A062163 this_sequence A030209 A138588 A143102

Adjacent sequences: A002726 A002727 A002728 this_sequence A002730 A002731 A002732

nonn,easy,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

Entry revised by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Max Alekseyev, Jun 20 2007