Takeuchi number

The nth Takeuchi numberMathworldPlanetmath Tn is the value of the functionMathworldPlanetmath T(n,0,n+1) which measures how many times the Takeuchi function t(x,y,z) has to call itself to give the answer starting with x=n,y=0,z=n+1. For example, the second Takeuchi number is 4, since t(2,0,3) 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 formulaMathworldPlanetmathPlanetmath for the asymptotic growth of the Takeuchi numbers:


, where c is the Takeuchi-Prellberg constant (approximately 2.2394331), Bn is the nth Bernoulli numberDlmfDlmfMathworldPlanetmathPlanetmath and W(x) is Lambert’s W function.


Title Takeuchi number
Canonical name TakeuchiNumber
Date of creation 2013-03-22 17:33:09
Last modified on 2013-03-22 17:33:09
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 5
Author PrimeFan (13766)
Entry type Definition
Classification msc 05A16