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

1,2

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.

Links

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

Example

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.

Mathematica

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

Crossrefs

Cf. A342042.

Sequence in context: A144113 A370329 A304038 * A301983 A165416 A222818

Adjacent sequences: A366110 A366111 A366112 * A366114 A366115 A366116

Keyword

nonn

Author

Eric Angelini, Sep 29 2023

Extensions

Data corrected by Giorgos Kalogeropoulos, Oct 03 2023

Status

approved