A002729 Number of equivalence classes of binary sequences of period n.

1, 2, 3, 4, 6, 6, 13, 10, 24, 22, 45, 30, 158, 74, 245, 368, 693, 522, 2637, 1610, 7386, 8868, 19401, 16770, 94484, 67562, 216275, 277534, 815558, 662370, 4500267, 2311470, 8466189, 13045108, 31593285, 40937606, 159772176, 103197490, 401913697

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

From Pab Ter (pabrlos2(AT)yahoo.com), Jan 24 2006: (Start)

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_{t=0..p-1} Sum_{k=1..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_{u=0..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)).

(End)

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

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.

References

• D. Z. Dokovic, I. Kotsireas, D. Recoskie, J. Sawada, Charm bracelets and their application to the construction of periodic Golay pairs, Discrete Applied Mathematics, Volume 188, 19 June 2015, Pages 32-40.
• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..50
• CombOS - Combinatorial Object Server, Generate necklaces, Lyndon words, chord diagrams, and relatives.
• D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
• M. Engelhardt, The N queens problem.
• R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
• Index entries for sequences related to necklaces

Crossrefs

Row 2 of A285548.

Cf. A002730.

Programs

Maple

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)

Mathematica

max = 38; m[k_, n_] := (s = 1; Do[ If[ Mod[s, n] == 0, Return[e], s = s + k^e ] , {e, 1, max}]); c[k_, t_, n_] := Sum[ m[k, n/GCD[n, u*(k-1) + t]]^(-1), {u, 0, n-1}]; a[n_] := (s = 0; Do[ If[ GCD[n, k] == 1 , s = s + 2^c[k, t, n]] , {k, 1, n-1}, {t, 0, n-1}]; s/(n*EulerPhi[n])); a[0] = 1; a[1] = 2; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Dec 06 2011, after Maple *)

Formula

Reference gives formula.

Extensions

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

Entry revised by N. J. A. Sloane at the suggestion of Max Alekseyev, Jun 20 2007