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The  th Takeuchi number  is the value of the function
 which measures how many times the Takeuchi function
 has to call itself to give the answer starting with
 . For example, the second Takeuchi number is 4, since
 requires four recursions to obtain the answer 2. The first few Takeuchi numbers are 0, 1, 4, 14, 53, 223, 1034, 5221, 28437, listed in A000651 of Sloane's OEIS. Prellberg gives a formula for the asymptotic growth of the Takeuchi numbers:
, where  is the Takeuchi-Prellberg constant (approximately 2.2394331),  is the  th Bernoulli number and  is Lambert's  function.
- 1
- Steven R. Finch Mathematical Constants New York: Cambridge University Press (2003): 321
- 2
- T. Prellberg, ``On the asymptotics of Takeuchi numbers'', Symbolic computation, number theory, special functions, physics and combinatorics, Dordrecht: Kluwer Acad. Publ. (2001): 231 - 242.
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