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Grossman's Constant

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GrossmansConstant

Grossman's constant is the value x=c for which the sequence defined by a_0=1, a_1=x>=0, and

a_n=(a_(n-2))/(1+a_(n-1))
(1)

for n>=2 converges. The first few values are

a_2=1/(1+x)
(2)
a_3=(x(1+x))/(2+x)
(3)
a_4=(2+x)/((1+x)(2+2x+x^2))
(4)
a_5=(x(1+x)^2(2+2x+x^2))/((2+x)(4+5x+3x^2+x^3)).
(5)

Janssen and Tjaden (1987) showed that this sequence converges for exactly one value x=c, where c=0.73733830336928... (OEIS A085835), confirming Grossman's conjecture. Finch (2003, p. 430) notes that no easily available expression in well-known constants or functions is known. The plot above shows the first few iterations of a_n for n=1 to 30, with odd n shown in red and even n shown in blue, for x ranging from 0 to 1. As can be seen, the solutions alternate by parity. For each fixed 0<=x<c, 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

x=[1+F(x)]F(F(x)),
(6)

where F:[0,infty)->[0,infty) is continuous and assigns to the initial value a_0=x the unique initial value a_1=F(x) for which the sequence converges. Thus c=F(1) (Finch 2003, pp. 429-430).

See also

Foias Constant

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References

Ewing, J. and Foias, C. "An Interesting Serendipitous Real Number." In Finite versus Infinite: Contributions to an Eternal Dilemma (Ed. C. S. Calude and G. Păun). London, England: 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." https://www.numericana.com/answer/constants.htm#grossman.Nyerges, G. "The Solution of the Functional Equation x=(1+F(x))F^2(x)." 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."

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Grossman's Constant

Cite this as:

Weisstein, Eric W. "Grossman's Constant." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GrossmansConstant.html

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