A366113 A fractal sequence. Pick any term t: exactly one of the t terms following t is larger than t.

1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 5, 1, 2, 1, 3, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 3, 1, 2, 8, 1, 2, 1, 3, 1, 2, 4, 9, 1, 2, 1, 3, 1, 2, 4, 1, 10, 1, 2, 1, 3, 1, 2, 4, 1, 2, 11, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 12, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 5, 13, 1, 2, 1, 3, 1, 2, 4, 1, 2, 1, 5, 1, 14

Offset: 1

Author

Eric Angelini, Sep 29 2023

Keywords

• nonn

Comments

Starting with a(1) = 1, this is the lexicographically earliest sequence with this property.

This is how the sequence was built:

1) start with the Natural numbers and increase the gaps between them by one unit (first gap is zero):

S = 1,2,,3,,_,4,,,_,5,,,_,,6,,_,,,_,7,,,_,,,_,8, …

2) fill the empty spaces with the successive starting chunks of the sequence S itself:

S = 1,2,,3,,_,4,,,_,5,,,_,,6,,_,,,_,7,,,_,,,_,8,_, …

S = 1,2,1,3,,,4,,,_,5,,,_,,6,,_,,,_,7,,,_,,,_,8,_, …

S = 1,2,1,3,1,2,4,,,_,5,,,_,,6,,_,,,_,7,,,_,,,_,8,_, …

S = 1,2,1,3,1,2,4,1,2,1,5,,,_,,6,,_,,,_,7,,,_,,,_,8,_, …

S = 1,2,1,3,1,2,4,1,2,1,5,1,2,1,3,6,,,_,,,7,,,_,,,_,8,_, …

S = 1,2,1,3,1,2,4,1,2,1,5,1,2,1,3,6,1,2,1,3,1,7,,,_,,,_,8,_, …

S = 1,2,1,3,1,2,4,1,2,1,5,1,2,1,3,6,1,2,1,3,1,7,1,2,1,3,1,2,8,_, … Etc.

Examples

			One term out of 1 following a(1) = 1 is larger than 1: this is 2.
One term out of 2 following a(2) = 2 is larger than 2: this is 3.
One term out of 1 following a(3) = 1 is larger than 1: this is 2.
One term out of 3 following a(4) = 3 is larger than 3: this is 4.
One term out of 1 following a(5) = 1 is larger than 1: this is 2.
One term out of 2 following a(6) = 2 is larger than 2: this is 4.
One term out of 4 following a(7) = 4 is larger than 4: this is 5.
One term out of 1 following a(8) = 1 is larger than 1: this is 2.
One term out of 2 following a(9) = 2 is larger than 2: this is 5. Etc.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A342042.

Programs

Mathematica

k=1;Nest[(k++;Join[#,#[[;;k-2]],{k}])&,{1},13] (* Giorgos Kalogeropoulos, Oct 03 2023 *)

Extensions

Data corrected by Giorgos Kalogeropoulos, Oct 03 2023