A007148 Number of self-complementary 2-colored bracelets (turnover necklaces) with 2n beads. (Formerly M0774)

1, 2, 3, 6, 10, 20, 37, 74, 143, 284, 559, 1114, 2206, 4394, 8740, 17418, 34696, 69194, 137971, 275280, 549258, 1096286, 2188333, 4369162, 8724154, 17422652, 34797199, 69505908, 138845926, 277383872, 554189329, 1107297290, 2212558942

Offset

1,2

References

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 = 1..1000

E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations Pacific J. Math., 110 (1984), 203-221.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]

Index entries for sequences related to bracelets

Formula

a(n) = 2^(n-2) + (1/(4n)) * Sum_{d|n} phi(2d)*2^(n/d). - N. J. A. Sloane, Sep 25 2012

a(n) = (1/2)*(A000079(n-1) + A000013(n)).

Maple

# see A245558
L := proc(n, k)
    local a, j ;
    a := 0 ;
    for j in numtheory[divisors](igcd(n, k)) do
        a := a+numtheory[mobius](j)*binomial(n/j, k/j) ;
    end do:
    a/n ;
end proc:
A007148 := proc(n)
    local a, k, l;
    a := 0 ;
    for k from 1 to n do
        for l in numtheory[divisors](igcd(n, k)) do
            a := a+L(n/l, k/l)*ceil(k/2/l) ;
        end do:
    end do:
    a;
end proc:
seq(A007148(n), n=1..20) ; # R. J. Mathar, Jul 23 2017

Mathematica

a[n_] := (1/2)*(2^(n-1) + Total[ EulerPhi[2*#]*2^(n/#) &  /@ Divisors[n]]/(2*n)); Table[ a[n], {n, 1, 33}] (* Jean-François Alcover, Oct 25 2011 *)

Prog

(PARI) a(n)= (1/2) *(2^(n-1)+sumdiv(n, k, eulerphi(2*k)*2^(n/k))/(2*n))
(Python)
from sympy import divisors, totient
def a(n):
    if n==1: return 1
    return 2**(n - 2) + sum(totient(2*d)*2**(n//d) for d in divisors(n))//(4*n)
print([a(n) for n in range(1, 31)]) # Indranil Ghosh, Jul 24 2017

Crossrefs

Cf. A000013, A000079, A007147.

Different from, but easily confused with, A045690 and A093371.

Sequence in context: A164047 A158291 A045690 * A093371 A339153 A003214

Adjacent sequences: A007145 A007146 A007147 * A007149 A007150 A007151

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Description corrected by Christian G. Bower

Status

approved