OFFSET
0,3
COMMENTS
"Even" orbits of binary necklaces of length 2n under group D_n X S_2.
REFERENCES
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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
Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Maple code for this and related sequences
FORMULA
a(0)=1, a(n) = (A000011(2*n) + A000011(n) + 4^(n/2-1) - 2^(n/2-1))/2 if n is even, a(n) = A000011(2*n)/2 if n is odd. - Randall L Rathbun, Jan 11 2002
MAPLE
with(numtheory):
b:= proc(n) option remember;
`if`(n=0, 1, 2^(floor(n/2)-1)
+add(phi(2*d) *2^(n/d), d=divisors(n))/(4*n))
end:
a:= n-> `if`(n=0, 1, `if`(irem(n, 2)=0,
(b(2*n) +b(n) +4^(n/2-1) -2^(n/2-1))/2, b(2*n)/2)):
seq(a(n), n=0..30); # Alois P. Heinz, Mar 25 2012
MATHEMATICA
a[0] = 1; a11[n_] := Fold[#1 + EulerPhi[2*#2]*(2^(n/#2)/(2*n)) & , 2^Floor[n/2], Divisors[n]]/2; a[(n_)?EvenQ] := (a11[2*n] + a11[n] + 4^(n/2 - 1) - 2^(n/2 - 1))/2; a[(n_)?OddQ] := a11[2*n]/2; Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Sep 01 2011, after PARI prog. *)
PROG
(PARI) {A000206(n)=if(n==0, 1, if(n%2==0, (A000011(2*n)+A000011(n)+4^(n/2-1)-2^(n/2-1))/2, A000011(2*n)/2))} \\ Randall L Rathbun, Jan 11 2002
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Randall L Rathbun, Jan 11 2002
STATUS
approved