A000206 Even sequences with period 2n.
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
Offset: 0
References
- 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).
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- N. J. A. Sloane, Maple code for this and related sequences
Programs
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Maple
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 -
Mathematica
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. *) -
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
Formula
a(0)=1, a(n) = (A000011(2*n) + A000011(n) + 4^(n/2-1) - 2^(n/2-1))/2 if n is even, a(n) = A000011(2*n)/2 if n is odd. - Randall L Rathbun, Jan 11 2002
Extensions
More terms from Randall L Rathbun, Jan 11 2002
Comments