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Shallit Constant

Define $f(x_1,x_2,\ldots,x_n)$ with $x_i$ Positive as

\begin{displaymath}
f(x_1, x_2, \ldots, x_n)\equiv \sum_{i=1}^n x_i+\sum_{1\leq i\leq k\leq n} \prod_{j=i}^k {1\over x_j}.
\end{displaymath}

Then

\begin{displaymath}
\min f = 3n - C + o(1)
\end{displaymath}

as $n$ increases, where the Shallit constant is

\begin{displaymath}
C = 1.369451403937\dots
\end{displaymath}

(Shallit 1995). In their solution, Grosjean and De Meyer (quoted in Shallit 1995) reduced the complexity of the problem.


References

MacLeod, A. http://www.mathsoft.com/asolve/constant/shapiro/macleod.html.

Shallit, J. Solution by C. C. Grosjean and H. E. De Meyer. ``A Minimization Problem.'' Problem 94-15 in SIAM Review 37, 451-458, 1995.




© 1996-9 Eric W. Weisstein
1999-05-26