Triple FJS

[hv at crypt.org]

"Eric Angelini" <keynews.tv at skynet.be> wrote:
:Hello SeqFans,
:I'm working on "Fractal Jump Sequences" for the moment
:(search the OEIS with the string "FJS"). I have found this:
:
:(a)   1  1 3   1 1 3   1 3   1 1 3   1 1 3   1 3   1 1 1 3   3 ...
:(b)  1 13 1 131 3 1 131 1 313 1 1 133 1 3 111 1 313 3 1 1 133 1 ...
:     ----------------------------------------------------------
:(c)  1113113131131131311133131111313311133111113313131111313331 ...
:     ^
:- the succession of digits in (a) and (b) are the same;
:- when I "push" (a) into (b) I get (c);
:- ... and (c) has the same succession of digits as (a) and (b)
:      [except for the initial "1"].
:
:Does this ring a bell to someone ? Is it of interest ?
[snip]

It reminds me of a sequence discussed by Hofstadter [0], of the number
of triangular numbers between each successive pair of squares.

Formally, a(n) is the number of triangular numbers T such that
n^2 <= T < (n+1)^2, and the sequence starts:
  2, 1, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 2 ...
(Eg a(3) = 2 because 3^2 <= T < 4^2 permits T(4) = 10 and T(5) = 15,
and no other triangular number.) I'm not sure, but I think this is
A006338 - if so I'm surprised it doesn't mention the definition above,
since it was apparently submitted by Hofstadter himself.

Hugo

[0] Douglas Hofstadter, "Fluid Concepts and Creative Analogies",
<http://www.amazon.com/exec/obidos/tg/detail/-/0465024750>
Chapter 1: "to seek whence cometh a sequence".