The Hofstadter-Conway $10,000 sequence is the recursive sequence defined by the recurrence relation
 |
(1)
|
with 
. The first few values are 1, 1, 2, 2, 3, 4, 4, 4,
5, 6, ... (OEIS A004001; Wolfram 2002, pp. 129-130,
sequence (c)). Conway (1988) showed that 
and offered $100 for any threshold
beyond which 
, and $10,000 for the least such threshold.
With the convention that the inequality must hold for every 
, Mallows (1991) found 
.
The plots above show 
(left plot) and 
(right plot). Amazingly, 
reveals itself to consist of a series of increasingly
larger versions of the batrachion blancmange
function.
takes a value of 1/2 for 
of the form 
with 
, 2, .... More generally,
 |
(2)
|
and
 |
(3)
|
Pickover (1995) gives a table of analogous values of 
corresponding to different values of 
.
A related chaotic sequence is given by the recurrence equation
 |
(4)
|
with 
, which gives the sequence
1, 1, 2, 2, 2, 3, 4, 4, 4, 4, 5, 6, 7, 8, 8, 8, 8, 8, 8, ... (OEIS A055748;
Pinn 2000; Wolfram 2002, pp. 129-130,
sequence (g)).
Cloitre (2026) studied the alternating perturbation
 |
(5)
|
with 
and 
. Its first terms are 1, 1, 1, 3, 1, 3, 3, 3, 3, 5, 3,
5, 3, 5, 5, 5, ... (OEIS A399773). Cloitre
(2026) proved that it is well defined and that
 |
(6)
|
where 
is the real root of 
and 
denotes big-O notation.
."
Ch. 25 in 
." Comput. & Graphics 19, 611-615,
1995. Reprinted in