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 a prize of to the discoverer of a value of for which for . The prize was subsequently claimed by Mallows, after
adjustment to Conway's "intended" prize of (Schroeder 1991), who found .
The plots above show (left plot) and (right plot). Amazingly, reveals itself to consist of a series of increasingly
larger versions of the batrachionBlancmange
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 .
Bloom, D. M. "Newman-Conway Sequence." Solution to Problem 1459. Math. Mag.68, 400-401, 1995.Conolly,
B. W. "Meta-Fibonacci Sequences." In Fibonacci
and Lucas Numbers, and the Golden Section (Ed. S. Vajda). New York:
Halstead Press, pp. 127-138, 1989.Conway, J. "Some Crazy Sequences."
Lecture at AT&T Bell Labs, July 15, 1988.Guy, R. K. "Three
Sequences of Hofstadter." §E31 in Unsolved
Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 231-232,
1994.Kubo, T. and Vakil, R. "On Conway's Recursive Sequence."
Disc. Math.152, 225-252, 1996.Mallows, C. L. "Conway's
Challenge Sequence." Amer. Math. Monthly98, 5-20, 1991.Pickover,
C. A. "The Drums of Ulupu." In Mazes
for the Mind: Computers and the Unexpected. New York: St. Martin's Press,
1993.Pickover, C. A. "The Crying of Fractal Batrachion ."
Ch. 25 in Keys
to Infinity. New York: W. H. Freeman, pp. 183-191, 1995.Pickover,
C. A. "The Crying of Fractal Batrachion ." Comput. & Graphics19, 611-615,
1995. Reprinted in Chaos
and Fractals, A Computer Graphical Journey: Ten Year Compilation of Advanced Research
(Ed. C. A. Pickover). Amsterdam, Netherlands: Elsevier, pp. 127-131,
1998.Pinn, K. "A Chaotic Cousin of Conway's Recursive Sequence."
Exp. Math.9, 55-66, 2000.Schroeder, M. "John Horton
Conway's 'Death Bet.' " Fractals,
Chaos, Power Laws. New York: W. H. Freeman, pp. 57-59, 1991.Sloane,
N. J. A. Sequences A004001/M0276
and A055748 in "The On-Line Encyclopedia
of Integer Sequences."Wolfram, S. A
New Kind of Science. Champaign, IL: Wolfram Media, pp. 129-130,
2002.
. 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 a prize of 
to the discoverer of a value of 
for which 
for 
. The prize was subsequently claimed by Mallows, after
adjustment to Conway's "intended" prize of 
(Schroeder 1991), who found 
.
(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,
corresponding to different values of 
.
, 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)).
."
Ch. 25 in 
." Comput. & Graphics 19, 611-615,
1995. Reprinted in