LINKS
G. C. Greubel, <a href="/A102406/b102406_1.txt">Table of n, a(n) for n = 0..1000</a>
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G. C. Greubel, <a href="/A102406/b102406_1.txt">Table of n, a(n) for n = 0..1000</a>
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G. C. Greubel, <a href="/A102406/b102406_1.txt">Table of n, a(n) for n = 0..1000</a>
G.f.: [(1+z+z^2 - sqrt(1-2z2*z-5z5*z^2-2z2*z^3+z^4)])/[2z(2*z*(1+z)^2]).
Conjecture: (n+1)*a(n) +(-(n+-3)*a(n-1) +(-(7*n+-9)*a(n-2) +(-(7*n+-12)*a(n-3) -n*a(n-4) +(n-4)*a(n-5) = 0. - R. J. Mathar, Jan 04 2017
CoefficientList[Series[(1+x+x^2 -Sqrt[1-2*x-5*x^2-2*x^3+x^4])/(2*x*(1+x)^2), {x, 0, 40}], x] (* G. C. Greubel, Oct 31 2024 *)
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (1+x+x^2 -Sqrt(1-2*x-5*x^2-2*x^3+x^4))/(2*x*(1+x)^2) )); // G. C. Greubel, Oct 31 2024
(SageMath)
def A102406_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1+x+x^2 -sqrt(1-2*x-5*x^2-2*x^3+x^4))/(2*x*(1+x)^2) ).list()
A102406_list(30) # G. C. Greubel, Oct 31 2024
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Conjecture: (n+1)*a(n) +(-n+3)*a(n-1) +(-7*n+9)*a(n-2) +(-7*n+12)*a(n-3) -n*a(n-4) +(n-4)*a(n-5)=0. - R. J. Mathar, Jan 04 2017
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Number of Lukasiewicz Łukasiewicz paths of length n having no level steps at an even level. A Lukasiewicz Łukasiewicz path of length n is a path in the first quadrant from (0,0) to (n,0) using rise steps (1,k) for any positive integer k, level steps (1,0) and fall steps (1,-1) (see R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge Univ. Press, Cambridge, 1999, p. 223, Exercise 6.19w; the integers are the slopes of the steps). Example: a(3)=2 because we have UHD and U(2)DD, where U=(1,1), H=(1,0), D=(1,-1) and U(2)=(1,2). a(n)=A102404(n,0).
<a href="/index/Lu#Lukasiewicz">Index entries for sequences related to Łukasiewicz</a>
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