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A165407
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Expansion of 1/(1-x-x^3*c(x^3)), c(x) the g.f. of A000108.
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5
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1, 1, 1, 2, 3, 4, 7, 11, 16, 27, 43, 65, 108, 173, 267, 440, 707, 1105, 1812, 2917, 4597, 7514, 12111, 19196, 31307, 50503, 80380, 130883, 211263, 337284, 548547, 885831, 1417582, 2303413, 3720995, 5965622, 9686617, 15652239, 25130844, 40783083, 65913927
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OFFSET
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0,4
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COMMENTS
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Number of UF-equivalence classes of Łukasiewicz paths. Łukasiewicz paths are UF-equivalent iff the positions of pattern UF are identical in these paths. This also works for the pattern FU. - Sergey Kirgizov, Apr 08 2018
a(n) is the total number of lattice paths from (0,0) to (n-2*i,i) that do not go above the diagonal x=y using steps in {(1,0), (0,1)}. - Alois P. Heinz, Sep 20 2022
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LINKS
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FORMULA
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G.f.: 2/(1 - 2*x + sqrt(1-4*x^3)) = 1/(1-x-x^3/(1-x^3/(1-x^3/(1-x^3/(1-.... (continued fraction).
a(n) = Sum_{k=0..n} if(n<=3k, C((n+k)/2,k)*((3*k-n)/2 + 1)*(1+(-1)^(n-k) )/(2*(k+1)).
a(n) = Sum_{k=0..n+1} Fibonacci(n-k+1)*(0^k - A000108((k-2)/3)*(1+2*cos(2*Pi*(k-2)/3))/3).
(n+1)*a(n) = (n+1)*a(n-1) + (n+1)*a(n-2) +2*(2*n-7)*a(n-3) -2*(2*n-7)*a(n-4) -2*(2*n-7)*a(n-5). - R. J. Mathar, Nov 15 2011
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MAPLE
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b:= proc(x, y) option remember; `if`(y<=x, `if`(x=0, 1,
b(x-1, y)+`if`(y>0, b(x, y-1), 0)), 0)
end:
a:= n-> add(b(n-2*i, i), i=0..n/3):
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MATHEMATICA
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b[x_, y_]:= b[x, y]= If[y<=x, If[x==0, 1, b[x-1, y] +If[y>0, b[x, y-1], 0]], 0];
a[n_] := Sum[b[n-2*i, i], {i, 0, n/3}];
CoefficientList[Series[(Sqrt[1-4*x^3] -1+2*x)/(2*x*(1-x-x^2)), {x, 0, 50}], x] (* G. C. Greubel, Nov 09 2022 *)
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PROG
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(Magma) R<x>:=PowerSeriesRing(Rationals(), 50); Coefficients(R!( (Sqrt(1-4*x^3) -1+2*x)/(2*x*(1-x-x^2)) )); // G. C. Greubel, Nov 09 2022
(SageMath)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( 2/(1-2*x+sqrt(1-4*x^3)) ).list()
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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