A000116 - OEIS

%I M1156 N0440 #47 Mar 14 2024 05:20:00

%S 1,2,4,8,20,56,180,596,2068,7316,26272,95420,349716,1290872,4794088,

%T 17896832,67110932,252648992,954444608,3616828364,13743921632,

%U 52357746896,199911300472,764877836468,2932031358484,11258999739560,43303843861744,166799988689300

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Alois P. Heinz, <a href="/A000116/b000116.txt">Table of n, a(n) for n = 0..1000</a>

%H E. N. Gilbert and J. Riordan, <a href="http://projecteuclid.org/euclid.ijm/1255631587">Symmetry types of periodic sequences</a>, Illinois J. Math., 5 (1961), 657-665.

%F a(2*n) + a(n) = 2 * A000208(2*n); a(2*n+1) = 2 * A000208(2*n+1). - _Reinhard Zumkeller_, Jul 08 2013

%F a(n) ~ 4^(n-1) / n. - _Cedric Lorand_, Apr 18 2022

%p with(numtheory):

%p a:= n-> `if`(n=0, 1, add(phi(2*d)*2^(2*n/d), d=divisors(2*n))/(4*n)):

%p seq(a(n), n=0..20);  # _Alois P. Heinz_, Mar 25 2012

%t 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_ *)

%o (Haskell)

%o a000116 n = a000116_list !! n

%o a000116_list = bis a000013_list where bis (x:_:xs) = x : bis xs

%o -- _Reinhard Zumkeller_, Jul 08 2013

%Y Cf. A000013, A026119.

%K nonn,easy,nice

%O 0,2

%A _N. J. A. Sloane_

%E More terms from _David W. Wilson_, Jan 13 2000