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. [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
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
KEYWORD
nonn,nice,easy
AUTHOR
EXTENSIONS
Description corrected by Christian G. Bower
STATUS
approved