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Search: id:A000651
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| A000651 |
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Running time of Takeuchi function. |
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+0
2
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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; listen)
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OFFSET
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0,3
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REFERENCES
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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
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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
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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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PROGRAM
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(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
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Sequence in context: A017948 A112872 A162482 this_sequence A118896 A145211 A060898
Adjacent sequences: A000648 A000649 A000650 this_sequence A000652 A000653 A000654
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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