A110010 a(n) = n-F(F(F(F(n)))) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.

1, 2, 3, 4, 5, 6, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7, 6, 6, 7, 6, 7, 6, 6, 7, 6, 5, 6, 6, 7

Offset: 1

Author

Benoit Cloitre, Sep 02 2005

Keywords

• nonn
• easy

Comments

To built the sequence start from the infinite Fibonacci word b(k)=floor(k/phi)-floor((k-1)/phi) for k>=1 giving 0,1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,..... Then replace each 0 by the block {5,6,6} and each 1 by the block {7, 6, 6, 7, 6}. Append the initial string {1,2,3,4}.

References

• Benoit Cloitre, On properties of irrational numbers related to the floor function, in preparation, 2005.

Crossrefs

Cf. A005614 (infinite Fibonacci binary word), A120613.

Cf. sequences for a(n) = n-F^k(n): A003842 (k=1), A110006 (k=2), A110007 (k=3), A110011 (k=5).

Programs

PARI

F(x)=floor((1+sqrt(5))/2*floor((-1+sqrt(5))/2*x)); a(n)=n-F(F(F(F(n))))