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
KEYWORD
nonn
AUTHOR
Eric Angelini, Sep 29 2023
EXTENSIONS
Data corrected by Giorgos Kalogeropoulos, Oct 03 2023
STATUS
approved