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A000116
Number of even sequences with period 2n (bisection of A000013).
(Formerly M1156 N0440)
4
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
0,2
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
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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
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 *)
PROG
(Haskell)
a000116 n = a000116_list !! n
a000116_list = bis a000013_list where bis (x:_:xs) = x : bis xs
-- Reinhard Zumkeller, Jul 08 2013
CROSSREFS
Sequence in context: A123611 A082279 A113180 * A302862 A344490 A006407
KEYWORD
nonn,easy,nice
EXTENSIONS
More terms from David W. Wilson, Jan 13 2000
STATUS
approved