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A000651
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Running time of Takeuchi function.
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1
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0, 1, 4, 14, 53, 223, 1034, 5221, 28437, 165859, 1029803, 6772850, 46983238, 342509396, 2615606677, 20865444825, 173446634597, 1499111445237, 13445550920288, 124919896067530, 1200320663197275, 11910845573790488
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| P. Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, Arxiv preprint arXiv:1107.5490, 2011.
D. E. Knuth, personal communication.
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.
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.
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LINKS
| T. Prellberg, On the asymptotic analysis of a class of linear recurrences (slides).
T. Prellberg, On the Asymptotics of Takeuchi Numbers
Eric Weisstein's World of Mathematics, Takeuchi Number
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FORMULA
| A(z-z^2)/z-A(z)=1/(1-z)+z/(1-z+z^2). (Prellberg).
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).
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PROG
| (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)))
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CROSSREFS
| Sequence in context: A017948 A112872 A162482 * A192247 A118896 A145211
Adjacent sequences: A000648 A000649 A000650 * A000652 A000653 A000654
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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