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

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

References

• 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
• E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Crossrefs

Cf. A000013, A026119.

Programs

Haskell

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

Maple

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

Mathematica

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 *)

Formula

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

Extensions

More terms from David W. Wilson, Jan 13 2000