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%I M2372 N0940 #34 Apr 30 2022 12:18:27
%S 1,1,3,4,12,22,71,181,618,1957,6966,24367,89010,324766,1204815,
%T 4482400,16802826,63195016,238711285,904338163,3436380192,13089961012,
%U 49979421837,191221556269,733014218506,2814758323498,10825986453978,41700030726757,160842946895004
%N Even sequences with period 2n.
%C "Even" orbits of binary necklaces of length 2n under group D_n X S_2.
%D E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
%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="/A000206/b000206.txt">Table of n, a(n) for n = 0..1000</a>
%H N. J. A. Sloane, <a href="/A000013/a000013.txt">Maple code for this and related sequences</a>
%F 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
%p with(numtheory):
%p b:= proc(n) option remember;
%p `if`(n=0, 1, 2^(floor(n/2)-1)
%p +add(phi(2*d) *2^(n/d), d=divisors(n))/(4*n))
%p end:
%p a:= n-> `if`(n=0, 1, `if`(irem(n, 2)=0,
%p (b(2*n) +b(n) +4^(n/2-1) -2^(n/2-1))/2, b(2*n)/2)):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Mar 25 2012
%t 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. *)
%o (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
%Y Cf. A000011, A000013, A000208.
%K nonn,easy,nice
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Randall L Rathbun_, Jan 11 2002
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