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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

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.

FORMULA

a(n) = (A000013(2*n)+A000013(n))/2 if n even, A000013(2*n)/2 if n 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

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-Fran├žois 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

Cf. A000013, A000206.

Sequence in context: A090660 A288140 A287594 * A079154 A101716 A217477

Adjacent sequences:  A000205 A000206 A000207 * A000209 A000210 A000211

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Randall L. Rathbun, Jan 11 2002

STATUS

approved

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Last modified June 25 20:26 EDT 2017. Contains 288730 sequences.