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A230122
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1-Fibonacci lattice paths.
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0
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1, 1, 3, 8, 23, 67, 199, 600, 1834, 5674, 17743, 56011, 178301, 571812, 1845913, 5993985, 19565770, 64168531, 211343740, 698753053, 2318315786, 7716186315, 25757105801, 86208990248, 289253059765, 972729789813
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OFFSET
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1,3
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COMMENTS
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a(n) is the number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and horizontal steps H(s) = (s,0) for every positive integer s, such that H(s) is colored by means of F(s) colors, where F(s) is the s-th Fibonacci number.
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LINKS
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FORMULA
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G.f.: (1-2*z-z^2-sqrt((1-2*z-z^2)^2-4*z^2*(1-z-z^2)^2))/(2*z^2*(1-z-z^2)).
D-finite with recurrence: n*(n+2)*a(n) = (7*n^2 + 5*n - 1)*a(n-1) - (14*n^2 - 8*n + 1)*a(n-2) + (3*n^2 - 12*n + 23)*a(n-3) + (n+1)*(13*n - 38)*a(n-4) - 4*(n-4)*n*a(n-5) - 4*(n-5)*(n+1)*a(n-6). - Vaclav Kotesovec, Mar 15 2014
a(n) ~ sqrt(663+161*sqrt(17)) * ((3+sqrt(17))/2)^(n-3/2) / (sqrt(2*Pi) * n^(3/2)). - Vaclav Kotesovec, Mar 15 2014
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MATHEMATICA
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Rest[CoefficientList[Series[(1-2*x-x^2-Sqrt[(1-2*x-x^2)^2-4*x^2*(1-x-x^2)^2])/(2*x^2*(1-x-x^2)), {x, 0, 20}], x]] (* Vaclav Kotesovec, Mar 15 2014 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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