A110007 a(n)=n-floor(phi*floor(phi^-1*floor(phi*floor(phi^-1*floor(phi*floor(phi^-1*n)))))) where phi=(1+sqrt(5))/2.

1, 2, 3, 4, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5, 5, 4, 5, 4, 5, 5, 4, 5, 4, 4, 5, 4, 5

Offset

1,2

Comments

To build 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 {4,5,4} and each 1 by the block {5,5,4,5,4}. Append the initial string {1,2,3,4}.

References

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

Links

Table of n, a(n) for n=1..105.

Mathematica

Join[{1, 2, 3, 4}, Flatten[Table[Floor[k/GoldenRatio]-Floor[(k-1)/ GoldenRatio], {k, 30}]/.{0->{4, 5, 4}, 1->{5, 5, 4, 5, 4}}]] (* Harvey P. Dale, Dec 12 2017 *)

Prog

(PARI) a(n)=n-floor((1+sqrt(5))/2*floor((-1+sqrt(5))/2*floor((1+sqrt(5))/2*floor((-1+sqrt(5))/2*n))))

Crossrefs

Cf. A003842 (case a(n)=n-floor(phi*floor(phi^-1*n)), A005614 (infinite Fibonacci binary word).

Sequence in context: A036371 A036370 A005208 * A327715 A306574 A088527

Adjacent sequences: A110004 A110005 A110006 * A110008 A110009 A110010

Keyword

nonn

Author

Benoit Cloitre, Sep 02 2005

Status

approved