%I A000651
%S A000651 0,1,4,14,53,223,1034,5221,28437,165859,1029803,6772850,46983238,
%T A000651 342509396,2615606677,20865444825,173446634597,1499111445237,
%U A000651 13445550920288,124919896067530,1200320663197275,11910845573790488
%N A000651 Running time of Takeuchi function.
%D A000651 D. E. Knuth, personal communication.
%D A000651 V. Lifschitz, editor, Artificial intelligence and mathematical theory 
               of computation. Papers in honor of John McCarthy. Academic Press, 
               Inc., Boston, MA, 1991. See p. 215.
%D A000651 T. Prellberg, On the asymptotics of Takeuchi numbers, Symbolic computation, 
               number theory, special functions, physics and combinatorics, Kluwer 
               Acad. Publ., Dordrecht, 2001, pp. 231-242. MR 2002m:11016.
%H A000651 T. Prellberg, <a href="http://algo.inria.fr/seminars/sem02-03/prellberg1-slides.ps">
               On the asymptotic analysis of a class of linear recurrences</a> (slides).
%H A000651 T. Prellberg, <a href="http://arXiv.org/abs/math.CO/0005008">On the Asymptotics 
               of Takeuchi Numbers</a>
%H A000651 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               TakeuchiNumber.html">Takeuchi Number</a>
%F A000651 A(z-z^2)/z-A(z)=1/(1-z)+z/(1-z+z^2). (Prellberg).
%F A000651 Asymptotic growth: a(n) ~ C_T*B(n)*exp(1/2*W(n)^2), where B(n) are the 
               Bernoulli numbers, W(n) the Lambert W function and C_T = 2.2394331040...(Prellberg).
%o A000651 (PARI) a(n)=if(n<1,0,sum(k=1,n,(2*k)!/k!/(k+1)!)+sum(k=0,n-2,(2*binomial(n+k-1,
               k)-binomial(n+k,k))*a(n-1-k)))
%Y A000651 Sequence in context: A017948 A112872 A162482 this_sequence A118896 A145211 
               A060898
%Y A000651 Adjacent sequences: A000648 A000649 A000650 this_sequence A000652 A000653 
               A000654
%K A000651 nonn
%O A000651 0,3
%A A000651 N. J. A. Sloane (njas(AT)research.att.com).