The Takeuchi numbers are defined by for integers, where is the number of times "otherwise" is called
in a full recursive evaluation of the TAK function,
without reusing previously computed values.
for integers, where is a Catalan number and
an empty sum is zero (Prellberg 2001, p. 232). The values
of
for ,
1, ... are 0, 1, 4, 14, 53, 223, 1034, 5221, 28437, ... (OEIS A000651).
Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 321, 2003.Knuth,
D. E. "Textbook Examples of Recursion." In Artificial
Intelligence and Mathematical Theory of Computation, Papers in Honor of John McCarthy
(Ed. V. Lifschitz). Boston, MA: Academic Press, pp. 207-229, 1991. https://arxiv.org/abs/cs/9301113.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 (Ed. F. G. Garvan
and M. E. H. Ismail). Dordrecht, Netherlands: Kluwer, pp. 231-242,
2001. https://doi.org/10.1007/978-1-4613-0257-5_14.Sloane,
N. J. A. Sequence A000651 in "The
On-Line Encyclopedia of Integer Sequences."
for integers 
, where 
is the number of times "otherwise" is called
in a full recursive evaluation of the TAK function,
without reusing previously computed values.
,
a recurrence relation for 
is
![T_n=sum_(k=0)^(n-2)[2(n+k-1; k)-(n+k; k)]T_(n-k-1)+sum_(k=1)^nC_k](/images/equations/TakeuchiNumber/NumberedEquation1.svg)
, where 
is a Catalan number and
an empty sum is zero (Prellberg 2001, p. 232). The values
of 
for 
,
1, ... are 0, 1, 4, 14, 53, 223, 1034, 5221, 28437, ... (OEIS A000651).
