A000208 -id:A000208 - cp's OEIS frontend

A000116 Number of even sequences with period 2n (bisection of A000013).

1, 2, 4, 8, 20, 56, 180, 596, 2068, 7316, 26272, 95420, 349716, 1290872, 4794088, 17896832, 67110932, 252648992, 954444608, 3616828364, 13743921632, 52357746896, 199911300472, 764877836468, 2932031358484, 11258999739560, 43303843861744, 166799988689300

Offset: 0

N. J. A. Sloane

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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Cf. A000013, A026119.

a000116 n = a000116_list !! n
a000116_list = bis a000013_list where bis (x:_:xs) = x : bis xs
-- Reinhard Zumkeller, Jul 08 2013

with(numtheory):
a:= n-> `if`(n=0, 1, add(phi(2*d)*2^(2*n/d), d=divisors(2*n))/(4*n)):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 25 2012

a[n_] := Sum[ EulerPhi[2d]*2^(2n/d), {d, Divisors[2n]}]/(4n); a[0]=1; Table[a[n], {n, 0, 27}] (* Jean-François Alcover, Sep 13 2012, after Alois P. Heinz *)

a(2*n) + a(n) = 2 * A000208(2*n); a(2*n+1) = 2 * A000208(2*n+1). - Reinhard Zumkeller, Jul 08 2013

a(n) ~ 4^(n-1) / n. - Cedric Lorand, Apr 18 2022

More terms from David W. Wilson, Jan 13 2000

A000206 Even sequences with period 2n.

Original entry on oeis.org

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

N. J. A. Sloane

"Even" orbits of binary necklaces of length 2n under group D_n X S_2.

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).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Maple code for this and related sequences

Cf. A000011, A000013, A000208.

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

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. *)

{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

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

More terms from Randall L Rathbun, Jan 11 2002

cp's OEIS Frontend

A000116 Number of even sequences with period 2n (bisection of A000013).

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