

A000208


Number of even sequences with period 2n.
(Formerly M2377 N0943)


3



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
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OFFSET

0,3


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


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



FORMULA



EXAMPLE

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


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}] (* JeanFrançois Alcover, Sep 01 2011, after PARI prog. *)


PROG

(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


CROSSREFS



KEYWORD

nonn,easy,nice


AUTHOR



EXTENSIONS



STATUS

approved



