A280510 Index sequence of the Thue-Morse sequence (A010060) as a block-fractal sequence.

3, 3, 6, 6, 12, 12, 12, 12, 24, 24, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 48, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 96, 192

Offset: 1

Author

Clark Kimberling, Feb 10 2017

Keywords

• nonn
• easy

Comments

The index sequence (a(n)) of a block-fractal sequence (s(n)) is defined by a(n) = least k > 0 such that (s(k), s(k+1), ..., s(k+n)) = (s(0), s(1), ..., s(n)).

Examples

			A010060 = (0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0,,...) = (s(0), s(1), ... ).
(initial block #1) = (0) first repeats at s(3), so that a(1) = 3;
(initial block #2) = (0,1) first repeats at s(3), so that a(2) = 3;
(initial block #3) = (0,1,1) first repeats at s(6), so that a(3) = 6.
		

Crossrefs

Cf. A010060, A280511, A280512.

Programs

Mathematica

seq = Table[Mod[Length[FixedPointList[BitAnd[#, # - 1] &, n]], 2], {n, 0, 400}]  (* A010060 *)
seq = StringJoin[Map[ToString, seq]]
u = -1 + Most[Flatten[Rest[Reap[NestWhile[# + 1 &, 1,
Sow[First[Last[StringPosition[seq, StringTake[seq, #], 2]]]] >
1 &]]]]] (* A280510 *)  (* Peter J. C. Moses, Jan 05 2017 *)

Formula

a(n) = 3*A053644(n-1) for n >= 2.