A134299 Maximal length of a sequence such that v(0)=n, v(k+2) = v(k)-v(k+1), v(k) >= 0.

4, 5, 6, 5, 7, 6, 5, 8, 6, 7, 6, 6, 9, 6, 7, 8, 6, 7, 6, 7, 10, 6, 7, 8, 7, 9, 6, 7, 8, 7, 7, 8, 7, 11, 7, 7, 8, 7, 9, 8, 7, 10, 7, 7, 8, 7, 9, 8, 7, 8, 7, 9, 8, 7, 12, 8, 7, 8, 7, 9, 8, 7, 10, 8, 9, 8, 7, 11, 8, 7, 8, 8, 9, 8, 7, 10, 8, 9, 8, 8, 9, 8, 7, 10, 8, 9, 8, 8, 13, 8, 9, 8, 8, 9, 8, 8, 10, 8, 9, 8

Offset: 1

Author

M. F. Hasler, Oct 18 2007

Keywords

• nonn

Comments

This is also the maximal index at which n can occur in a Fibonacci-like sequence u(k+2) = u(k)+u(k+1) of nonnegative numbers.

A sequence of this length is obtained for v(0) = n, v(1) = A019446(n) = ceiling(n/tau) or A060143(n) = floor(n/tau).

Examples

			a(2007)=11 since there is no such sequence longer than v = (2007, 1240, 767, 473, 294, 179, 115, 64, 51, 13, 38).
		

Links

• Eric Angelini, Max Alekseyev and M. F. Hasler, Longest Lucas sequence from start to n, postings to Sequence Fans mailing list, Oct 18, 2007.

Crossrefs

Cf. A101803, A019446, A060143.

Programs

PARI

A134299( goal, mi=0, mx=0, new=0 ) = { for( j=mi,goal, a=[goal,new=j]; while( mi<=new=a[ #a-1]-new, a=concat(a,new)); if( #a>mx, mx=#a)); mx }