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A054761
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Number of positive braids with n crossings of 5 strands.
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3
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1, 4, 13, 37, 99, 254, 636, 1567, 3822, 9261, 22346, 53773, 129174, 309958, 743228, 1781330, 4268166, 10224885, 24492034, 58662298, 140498877, 336491169, 805872377, 1929983778, 4622083068, 11069289411, 26509431448, 63486333364
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OFFSET
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0,2
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COMMENTS
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The (n+1)-strand braid group B_{n+1} has n generators s_1,...,s_n with relations s_i s_k s_i = s_k s_i s_k for k=i+1, s_i s_k = s_k s_i for k>i+1. The elements are the isotopy classes of (n+1) free strands that are planarily "mixed" (s_i corresponds to the operation of crossing the i-th strand under the (i+1)-th strand).
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (4, -3, -3, 2, 0, 2, 0, 0, 0, -1).
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FORMULA
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G.f.: 1/(1-4*x+3*x^2+3*x^3-2*x^4-2*x^6+x^10). - Ignat Soroko, Sep 30 2010
a(n) = 4*a(n-1) - 3*a(n-2) - 3*a(n-3) + 2*a(n-4) + 2*a(n-6) - a(n-10). - Wesley Ivan Hurt, May 12 2023
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MATHEMATICA
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CoefficientList[Series[1/(1-4x+3x^2+3x^3-2x^4-2x^6+x^10), {x, 0, 30}], x] (* or *) LinearRecurrence[{4, -3, -3, 2, 0, 2, 0, 0, 0, -1}, {1, 4, 13, 37, 99, 254, 636, 1567, 3822, 9261}, 30] (* Harvey P. Dale, May 09 2017 *)
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PROG
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(PARI) x='x+O('x^30); Vec(1/(1-4*x+3*x^2+3*x^3-2*x^4-2*x^6+x^10)) \\ G. C. Greubel, Jan 17 2018
(Magma) I:=[1, 4, 13, 37, 99, 254, 636, 1567, 3822, 9261]; [n le 10 select I[n] else 4*Self(n-1) - 3*Self(n-2) -3*Self(n-3) +2*Self(n-4) +2*Self(n-6) -Self(n-10): n in [1..30]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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Serge Burckel (burckel(AT)univ-reunion.fr), Apr 27 2000
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EXTENSIONS
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STATUS
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approved
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