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A000206
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Even sequences with period 2n.
(Formerly M2372 N0940)
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2
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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
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OFFSET
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0,3
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COMMENTS
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"Even" orbits of binary necklaces of length 2n under group D_n X S_2.
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REFERENCES
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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).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..1000
N. J. A. Sloane, Maple code for this and related sequences
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FORMULA
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a(0)=1, a(n) = (A000011(2*n) + A000011(n) + 4^(n/2-1) - 2^(n/2-1))/2 if n even, a(n) = A000011(2*n)/2 if n odd. - Randall L. Rathbun, Jan 11 2002
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MAPLE
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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
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MATHEMATICA
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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. *)
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PROG
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(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
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CROSSREFS
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Cf. A000011, A000013, A000208.
Sequence in context: A129922 A005221 A243391 * A240737 A075223 A071332
Adjacent sequences: A000203 A000204 A000205 * A000207 A000208 A000209
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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More terms from Randall L. Rathbun, Jan 11 2002
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STATUS
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approved
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