Takeuchi Number

Let T(x,y,z) be the number of times "otherwise" is called in the TAK function, then the Takeuchi numbers are defined by T_n(n,0,n+1).

A recursive formula for T_n is given by

 T_n=sum_(k=0)^(n-2)[2(n+k-1; k)-(n+k; k)]t_(n-k-1)+sum_(k=1)^nC_n,

where C_n is a Catalan number. The values for n=0, 1, ... are 0, 1, 4, 14, 53, 223, 1034, 5221, 28437, ... (OEIS A000651).

See also

TAK Function, Takeuchi-Prellberg Constant

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Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 321, 2003.Knuth, D. E. "Textbook Examples of Recursion." Artificial Intelligence and Mathematical Theory of Computation, Papers in Honor of John McCarthy (Ed. V. Lifschitz). Boston, MA: Academic Press, pp. 207-229, 1990.Lifschitz, V. (Ed.). Artificial Intelligence and Mathematical Theory of Computation. Papers in Honor of John McCarthy. Boston, MA: Academic Press, p. 215, 1991.Prellberg, T. "On the Asymptotics of Takeuchi Numbers." In Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Dordrecht, Netherlands: Kluwer, pp. 231-242, 2001.Sloane, N. J. A. Sequence A000651 in "The On-Line Encyclopedia of Integer Sequences."

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Takeuchi Number

Cite this as:

Weisstein, Eric W. "Takeuchi Number." From MathWorld--A Wolfram Web Resource.

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