A110006 -id:A110006 - cp's OEIS frontend

A110007 a(n) = n-F(F(F(n))) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and 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

Benoit Cloitre, Sep 02 2005

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}.

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

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

Cf. sequences for a(n) = n-F^k(n): A003842 (k=1), A110006 (k=2), A110010 (k=4), A110011 (k=5).

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 *)

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

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.

Original entry on oeis.org

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

Benoit Cloitre, Sep 02 2005

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}.

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

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).

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

A110011 a(n) = n-F(F(F(F(F(n))))) = n-F^5(n) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 8, 8, 9, 8, 7, 8, 8, 7, 8, 7, 8

Offset: 1

Benoit Cloitre, Sep 02 2005

To built the sequence start from the infinite Fibonacci word b(k)=floor(k/phi)-floor((k-1)/phi) for k>=2 giving 1,0,1,1,0,1,0,1,1,0,1,1,0,1,0,1,1,..... Then replace each 0 by the block {9,8,7,8,8,7,8,7,8,8,7,8,8} and each 1 by the block {9,8,7,8,8,7,8,7,8,8,7,8,8,7,8,7,8,8,7,8,8}. Append the initial string {1,2,3,4,5,6,7,8,8,7,8,8,7,8,7,8,8,7,8,8}.

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

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), A110010 (k=4).

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

cp's OEIS Frontend

A110007 a(n) = n-F(F(F(n))) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.

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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.

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A110011 a(n) = n-F(F(F(F(F(n))))) = n-F^5(n) where F(x)=A120613(x)=floor(phi*floor(x/phi)) and phi=(1+sqrt(5))/2.

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