A054761 - OEIS

%I #24 May 12 2023 14:57:06

%S 1,4,13,37,99,254,636,1567,3822,9261,22346,53773,129174,309958,743228,

%T 1781330,4268166,10224885,24492034,58662298,140498877,336491169,

%U 805872377,1929983778,4622083068,11069289411,26509431448,63486333364

%N Number of positive braids with n crossings of 5 strands.

%C 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).

%H G. C. Greubel, <a href="/A054761/b054761.txt">Table of n, a(n) for n = 0..1000</a>

%H F. A. Garside, <a href="https://doi.org/10.1093/qmath/20.1.235">The braid group and other groups</a>, Quart. Journal Math. Oxford,  Volume 20, Issue 1, 1 January 1969, Pages 235-254.

%H Kyoji Saito, <a href="https://doi.org/10.3792/pjaa.84.179">Growth functions associated with Artin monoids of finite type</a>, Proc. Japan Acad. Ser. A Math. Sci., 84 (2008), no. 10, 179-183. [_Ignat Soroko_, Sep 30 2010]

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4, -3, -3, 2, 0, 2, 0, 0, 0, -1).

%F 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

%F 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

%t 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 *)

%o (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

%o (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]];

%Y Cf. A000071, A054480.

%K nonn

%O 0,2

%A Serge Burckel (burckel(AT)univ-reunion.fr), Apr 27 2000

%E More terms from _Ignat Soroko_, Sep 30 2010