Janssen and Tjaden (1987) showed that this sequence converges for exactly one value ,
where
(OEIS A085835), confirming Grossman's conjecture.
However, no analytic form is known for this constant, either as the root of a function
or as a combination of other constants. The plot above shows the first few iterations
of
for
to 30, with odd
shown in red and even shown in blue, for ranging from 0 to 1. As can be seen, the solutions alternate
by parity. For each fixed , the red values go to 0, while the blue values go to
some positive number.
Nyerges (2000) has generalized the recurrence to the functional equation
Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite
versus Infinite: Contributions to an Eternal Dilemma (Ed. C. Caluse
and G. Păun). London: Springer-Verlag, pp. 119-126, 2000.Finch,
S. R. "Grossman's Constant." §6.4 in Mathematical
Constants. Cambridge, England: Cambridge University Press, pp. 429-430,
2003.Grossman, J. W. "Problem 86-2." Math. Intel.8,
31, 1986.Janssen, A. J. E. M. and Tjaden, D. L. A.
Solution to Problem 86-2. Math. Intel.9, 40-43, 1987.Michon,
G. P. "Final Answers: Numerical Constants." http://home.att.net/~numericana/answer/constants.htm#grossman.Nyerges,
G. "The Solution of the Functional Equation ." Preprint, Oct. 19, 2000. http://eent3.sbu.ac.uk/Staff/nyergeg/www/etc/fneq.pdf.Sloane,
N. J. A. Sequence A085835 in "The
On-Line Encyclopedia of Integer Sequences."
,
,
and
.
The first few values are
,
where 
(OEIS A085835), confirming Grossman's conjecture.
However, no analytic form is known for this constant, either as the root of a function
or as a combination of other constants. The plot above shows the first few iterations
of 
for 
to 30, with odd 
shown in red and even 
shown in blue, for 
ranging from 0 to 1. As can be seen, the solutions alternate
by parity. For each fixed 
, the red values go to 0, while the blue values go to
some positive number.
![x=[1+F(x)]F^2(x).](/images/equations/GrossmansConstant/NumberedEquation2.svg)
." Preprint, Oct. 19, 2000.