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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 (list; graph; refs; listen; history; text; internal format)
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
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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
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}] (* Jean-François Alcover, Sep 01 2011, after PARI prog. *)
PROG
(PARI) {A000208(n)=if(n%2==0, (A000013(2*n)+A000013(n))/2, A000013(2*n)/2)}
(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
CROSSREFS
Sequence in context: A288140 A287594 A296271 * A079154 A101716 A360992
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Randall L Rathbun, Jan 11 2002
STATUS
approved

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Last modified December 8 13:46 EST 2023. Contains 367679 sequences. (Running on oeis4.)