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

1, 3, 8, 19, 44, 102, 237, 551, 1281, 2978, 6923, 16094, 37414, 86977, 202197, 470051, 1092736, 2540303, 5905488, 13728594, 31915109, 74193627, 172479257, 400965626, 932131991, 2166943978, 5037533578, 11710844769, 27224411129, 63289077427

Offset: 0

Author

D n Verma, Aug 16 2004

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,1).

Crossrefs

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

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

Programs

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

Maple

a:= n-> (<<1|1|2>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1$2]:
seq(a(n), n=0..50);  # Alois P. Heinz, Jul 24 2008

Mathematica

LinearRecurrence[{3,-2,1},{1,3,8},30] (* Harvey P. Dale, Jul 10 2019 *)

Sage

@CachedFunction
def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
def A097550(n): return A095263(n) +A095263(n-2)
[A097550(n) for n in (0..30)] # G. C. Greubel, Apr 19 2021

Formula

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

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

a(n) = A095263(n) + A095263(n-2). - G. C. Greubel, Apr 19 2021

Extensions

More terms from Ryan Propper, Sep 27 2005