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A007148 Number of self-complementary 2-colored bracelets (turnover necklaces) with 2n beads.
(Formerly M0774)
8
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
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. [Cached copy, with permission, pdf format only]
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
Different from, but easily confused with, A045690 and A093371.
Sequence in context: A164047 A158291 A045690 * A093371 A339153 A003214
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Description corrected by Christian G. Bower
STATUS
approved

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Last modified May 23 10:34 EDT 2024. Contains 372760 sequences. (Running on oeis4.)