A383999 Sequence obtained by replacing 3-term subwords of A003849 by 0,1,2,3 as described in Comments.

1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2

Offset

1,2

Comments

The four 3-term subwords of A003849 are 0,0,1; 0,1,0; 1,0,0; 1,0,1. These are coded as 0,1,2,3, respectively, and then these numbers replace the corresponding subwords in A003849. In the resulting sequence, the positions of 0,1,2,3 are given by A003623, A003622, A035336, A101864, which are the x-Wythoff sequences for x = AB, AA, BA, BB, respectively. The corresponding limiting densities are sqrt(5)-2, (3-sqrt(5))/2, sqrt(5)-2, (7-3*sqrt(5))/2.

Links

Table of n, a(n) for n=1..86.

Example

Starting with A003849 = (0,1,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,...), the successive 3-term subwords are 0,1,0; 1,0,0; 0,0,1; 0,1,0; 1,0,1; ..., which code as 1,2,0,1,3,... .

Mathematica

Partition[Nest[Flatten[#/. {0->{0, 1}, 1->{0}}]&, {0}, 10], 3, 1]/.Thread[{{0, 0, 1}, {0, 1, 0}, {1, 0, 0}, {1, 0, 1}}->{0, 1, 2, 3}]  (* Peter J. C. Moses, May 22 2025 *)

Crossrefs

Cf. A003849, A003623, A003622, A035336, A101864, A035513, A381848.

Sequence in context: A369814 A239928 A114912 * A231723 A342720 A029274

Adjacent sequences: A383996 A383997 A383998 * A384000 A384001 A384002

Keyword

nonn

Author

Clark Kimberling, May 23 2025

Status

approved