A076895 a(1) = 1, a(n) = n - a(ceiling(n/2)).

1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 27, 28, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 37, 38, 38, 39, 39, 40, 41, 42, 42, 43, 43, 44, 44, 45, 46, 47, 47, 48, 48

Offset: 1

Author

Benoit Cloitre, Nov 26 2002

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A050292, A079411.

Programs

Mathematica

a[1] = 1; a[n_] := a[n] = n - a[Ceiling[n/2]]; Array[a, 100] (* Amiram Eldar, Sep 10 2020 *)

PARI

a(n)=if(n<2,1,n-a(ceil(n/2)))

Formula

a(n) is asymptotic to (2/3)*n.

a(n) = A050292(n-1) + (1+(1-2*(C(n)-F(n)))*(-1)^F(n))/2 where C(n) = ceiling(log_2(n)); F(n) = floor(log_2(n)) and A050292(n) (with A050292(0)=0) is the maximum cardinality of a double-free subset of {1, 2, ..., n}. So using Wang's asymptotic formula for A050292, a(n) = (2/3)*n + O(log_4(n)). Series expansion: (1/(x-1)) * ( 1/(x-1) + Sum_{i>=1} (-1)^i*( x^(2^i)/(x^(2^i)-1) - x^(2^(i-1)) ) ). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003

a(n+1) = b(n) with b(0)=0, b(2n) = -b(n) + 2n+1, b(2n+1) = -b(n) + 2n + 2 -[n==0]. Also a(n+1) = A050292(n) + A030301(n). - Ralf Stephan, Oct 28 2003

Extensions

Edited by Ralf Stephan, Sep 01 2004