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A097550 Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.

%I A097550 #31 Sep 08 2022 08:45:14
%S A097550 1,3,8,19,44,102,237,551,1281,2978,6923,16094,37414,86977,202197,
%T A097550 470051,1092736,2540303,5905488,13728594,31915109,74193627,172479257,
%U A097550 400965626,932131991,2166943978,5037533578,11710844769,27224411129,63289077427
%N A097550 Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.
%H A097550 Alois P. Heinz, <a href="/A097550/b097550.txt">Table of n, a(n) for n = 0..1000</a>
%H A097550 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,1).
%F A097550 G.f.: (1+x^2)/(1 - 3*x+ 2*x^2 - x^3).
%F A097550 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 A097550 a(n) = A095263(n) + A095263(n-2). - _G. C. Greubel_, Apr 19 2021
%p A097550 a:= n-> (<<1|1|2>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1$2]:
%p A097550 seq(a(n), n=0..50);  # _Alois P. Heinz_, Jul 24 2008
%t A097550 LinearRecurrence[{3,-2,1},{1,3,8},30] (* _Harvey P. Dale_, Jul 10 2019 *)
%o A097550 (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 A097550 (Sage)
%o A097550 @CachedFunction
%o A097550 def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
%o A097550 def A097550(n): return A095263(n) +A095263(n-2)
%o A097550 [A097550(n) for n in (0..30)] # _G. C. Greubel_, Apr 19 2021
%Y A097550 Cf. A097551, A097552, A097553, A097554, A097555, A097556.
%Y A097550 Cf. A095263, A135364, A136302, A136303, A136304, A136305, A137229, A137234, A137249.
%K A097550 nonn
%O A097550 0,2
%A A097550 _D n Verma_, Aug 16 2004
%E A097550 More terms from _Ryan Propper_, Sep 27 2005