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Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.
15

%I #31 Sep 08 2022 08:45:14

%S 1,3,8,19,44,102,237,551,1281,2978,6923,16094,37414,86977,202197,

%T 470051,1092736,2540303,5905488,13728594,31915109,74193627,172479257,

%U 400965626,932131991,2166943978,5037533578,11710844769,27224411129,63289077427

%N Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.

%H Alois P. Heinz, <a href="/A097550/b097550.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1).

%F G.f.: (1+x^2)/(1 - 3*x+ 2*x^2 - x^3).

%F a(n) = term (1,1) in the 1 X 3 matrix [1,1,2].[3,1,0; -2,0,1; 1,0,0]^n. - _Alois P. Heinz_, Jul 24 2008

%F a(n) = A095263(n) + A095263(n-2). - _G. C. Greubel_, Apr 19 2021

%p a:= n-> (<<1|1|2>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1$2]:

%p seq(a(n), n=0..50); # _Alois P. Heinz_, Jul 24 2008

%t LinearRecurrence[{3,-2,1},{1,3,8},30] (* _Harvey P. Dale_, Jul 10 2019 *)

%o (Magma) [n le 3 select Fibonacci(2*n) else 3*Self(n-1) -2*Self(n-2) +Self(n-3): n in [1..31]]; // _G. C. Greubel_, Apr 19 2021

%o (Sage)

%o @CachedFunction

%o def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )

%o def A097550(n): return A095263(n) +A095263(n-2)

%o [A097550(n) for n in (0..30)] # _G. C. Greubel_, Apr 19 2021

%Y Cf. A097551, A097552, A097553, A097554, A097555, A097556.

%Y Cf. A095263, A135364, A136302, A136303, A136304, A136305, A137229, A137234, A137249.

%K nonn

%O 0,2

%A _D n Verma_, Aug 16 2004

%E More terms from _Ryan Propper_, Sep 27 2005