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A352737
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Number of oriented two-component rational links (or two-bridge links) with crossing number n (a chiral pair is counted as two distinct ones).
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0
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2, 0, 4, 2, 10, 10, 30, 42, 102, 170, 374, 682, 1430, 2730, 5590, 10922, 22102, 43690, 87894, 174762, 350550, 699050, 1400150, 2796202, 5596502, 11184810, 22377814, 44739242, 89494870, 178956970, 357946710, 715827882, 1431721302, 2863311530, 5726754134, 11453246122
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OFFSET
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2,1
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COMMENTS
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The formula has been proved.
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REFERENCES
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Yuanan Diao, Michael Lee Finney, Dawn Ray. The number of oriented rational links with a given deficiency number, Journal of Knot Theory and its Ramifications, Vol 30, Number 9, 2021. 2150065_1-20. See Theorem 4.3 and its proof.
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LINKS
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FORMULA
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a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2).
G.f.: 2*x^2*(1 - x - 2*x^2 + x^3)/((1 + x)^(1 - 2*x)*(1 - 2*x^2)). - Stefano Spezia, Mar 31 2022
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EXAMPLE
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If n=2 there are two rational links, namely, the Hopf link pair, one with positive crossings and the other with negative crossings. There are no two-component rational links with crossing number 3.
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PROG
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(PARI) a(n) = (2^(n-2) + (-1)^n*2)/3 + ((-1)^n+1)*2^((n-4)/2); \\ Michel Marcus, Mar 31 2022
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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