A000206 Even sequences with period 2n. (Formerly M2372 N0940)

1, 1, 3, 4, 12, 22, 71, 181, 618, 1957, 6966, 24367, 89010, 324766, 1204815, 4482400, 16802826, 63195016, 238711285, 904338163, 3436380192, 13089961012, 49979421837, 191221556269, 733014218506, 2814758323498, 10825986453978, 41700030726757, 160842946895004

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

Cf. A000011, A000013, A000208.

Sequence in context: A129922 A005221 A243391 * A368031 A240737 A075223

Adjacent sequences: A000203 A000204 A000205 * A000207 A000208 A000209

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Randall L Rathbun, Jan 11 2002

Status

approved