A138967 Infinite Fibonacci word on the alphabet {1,2,3,4}.

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

Offset

1,2

Comments

Start with the infinite Fibonacci word A003849, which is 0100101001001010010... and replace each 0 by 1,2,3 and each 1 by 1,4.

(a(n)) is the unique fixed point of the morphism 1->12, 2->3, 3->14, 4->3, obtained by coding the overlapping 3-block morphism of the Fibonacci morphism according to 010<->1, 100<->2, 001<->3, 101<->4. - Michel Dekking, Sep 28 2017

Links

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

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

Formula

a(n) = 3 for n = 3, 8, 21, 55, ..., F(2*k), where k>1.

a(n) = 4 for n = 5, 13, 34, 89, ..., F(2*k+1), where k>1.

Let A(n)=floor(n*tau), B(n)=n+floor(n*tau); i.e., A and B are the lower and upper Wythoff sequences, A=A000201, B=A001950. a(n)=1 if n=A(A(k)) for some k; a(n)=2 if n=B(A(k)) for some k; a(n)=3 if n=A(B(k)) for some k; a(n)=4 if n=B(B(k)) for some k.

Crossrefs

Cf. A000201, A001950, A003849, A101864, A270788, A276757.

Keyword

nonn

Author

Clark Kimberling, Apr 04 2008

Status

approved

A255825 A self-generating sequence: a(n) = n for n < 5; a(5n) = a(n); if every 5th term (a(5), a(10), a(15),...) is deleted, this gives back the original sequence.

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

Offset

1,2

Comments

This is the m=5 analog of the m=10 variant A126616. Sequence A117943 is the m=3 analog with all terms decreased by 1.

Links

Table of n, a(n) for n=1..100.

Formula

a(n) = a(n/5) if n==0 (mod 5); a(n) = a(n - floor(n/5)) otherwise.

Prog

(PARI) a(n, m=5)=while(n>=m, if(n%m, n-=n\m, n\=m)); n \\ M. F. Hasler, Mar 07 2015

Crossrefs

Cf. A117943, A255824 - A255829, A126616.

Keyword

nonn

Author

M. F. Hasler, Mar 07 2015

Status

approved