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A097550
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Number of positive words of length n in the monoid Br_3 of positive braids on 4 strands.
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15
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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
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OFFSET
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0,2
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LINKS
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FORMULA
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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
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MAPLE
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a:= n-> (<<1|1|2>>. <<3|1|0>, <-2|0|1>, <1|0|0>>^n)[1$2]:
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MATHEMATICA
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LinearRecurrence[{3, -2, 1}, {1, 3, 8}, 30] (* Harvey P. Dale, Jul 10 2019 *)
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PROG
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(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
(Sage)
@CachedFunction
def A095263(n): return sum( binomial(n+j+2, 3*j+2) for j in (0..n//2) )
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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