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

Author

Clark Kimberling, May 23 2025

Keywords

• nonn

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.

Examples

			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,... .
		

Crossrefs

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

Programs

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 *)