A000208 Number of even sequences with period 2n.

1, 1, 3, 4, 12, 28, 94, 298, 1044, 3658, 13164, 47710, 174948, 645436, 2397342, 8948416, 33556500, 126324496, 477225962, 1808414182, 6871973952, 26178873448, 99955697946, 382438918234, 1466015854100, 5629499869780

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

These are binary sequences (sequences of 1's and 0's), and two sequences are considered the same if one can be transformed into the other by translation and/or exchanging 1 and 0. A periodic sequence can be represented by enclosing one period in parentheses (for example, (00011011)). Even sequences contain an even number of 1's and an even number of 0's. - Michael B. Porter, Dec 22 2019

Examples

			For n=2, the sequences of length 2n=4 are (0000), (0001), (0011), and (0101). The other 12 possibilities are equivalent - for example, the sequence (1001) is a translation of (0011), and the sequence (1101) is equivalent to (0001) by exchanging 1's and 0's and then translating. Since three of these have an even number of 1's, a(2) = 3. - _Michael B. Porter_, Dec 22 2019
		

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

• Vincenzo Librandi, 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, A000206.

Programs

Haskell

a000208 n = a000208_list !! n
a000208_list = map (`div` 2) $ concat $ transpose
   [zipWith (+) a000116_list $ bis a000116_list, bis $ tail a000116_list]
   where bis (x:_:xs) = x : bis xs
-- Reinhard Zumkeller, Jul 08 2013

Mathematica

a[0] = 1; a13[0] = 1; a13[n_] := Fold[#1 + EulerPhi[2*#2]*(2^(n/#2)/(2*n)) & , 0, Divisors[n]]; a[(n_)?OddQ] := (a13[2*(n + 1)] + a13[n + 1])/2; a[(n_)?EvenQ] := a13[2*(n + 1)]/2; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 01 2011, after PARI prog. *)

PARI

{A000208(n)=if(n%2==0,(A000013(2*n)+A000013(n))/2, A000013(2*n)/2)}

Formula

a(n) = (A000013(2*n) + A000013(n))/2 if n is even, A000013(2*n)/2 if n is odd. - Randall L Rathbun, Jan 11 2002

a(2*n) = (A000116(2*n) + A000116(n)) / 2; a(2*n+1) = A000116(2*n+1) / 2. - Reinhard Zumkeller, Jul 08 2013

Extensions

More terms from Randall L Rathbun, Jan 11 2002