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Partial sums ofA010006.So this sequence is the crystal ball sequence for the C_3 lattice - row 3 ofA142992.The lattice C_3 consists of all integer lattice points v = (a,b,c) in Z^3 such that a + b + c is even, equipped with the taxicab type norm ||v|| =(1/2)* (|a| + |b| + |c|).
Sum_{k = 1..n+1} 1/(k*a(k)*a(k+1)) = 1/(19 - 3/(27 - 60/(43 - 315/(67 -... -n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*3^2))))).
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R. Bacher, P. de la Harpe and B. Venkov, <a href= "http://archive.numdam.org/ARCHIVE/AIF/AIF_1999__49_3/AIF_1999__49_3_727_0SeriesSériesde croissance etseriessériesd'Ehrhartassocieesassociéesauxreseauxréseauxde racines</a>, C. R. Acad. Sci. Paris, 325 (Series 1) (1997), 1137-1142.
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E.g.f.: exp(x)*(1 + 18*x + 48*x^2/2! + 32*x^3/3!). Note that -T(6, i*sqrt(x)) = 1 + 18*x + 48*x^2 + 32*x^3, where T(n, x) denotes then-thChebyshev polynomial of the first kind. SeeA008310.(End)
E.g.f.: exp(x)*(1 + 18*x + 48*x^2/2! + 32*x^3/3!).Note that -T(6, i*sqrt(x)) = 1 + 18*x + 48*x^2 + 32*x^3, where T(n, x) denotes the Chebyshev polynomial of the first kind. SeeA008310.(End)
FromPeter Bala,Mar 11 2024: (Start)
Sum_{k = 1..n+1} 1/(k*a(k)*a(k+1))= 1/(19 - 3/(27 - 60/(43 - 315/(67 -... -n^2*(4*n^2 - 1)/((2*n + 1)^2 + 2*3^2 ))))).-_PeterBala_,Mar112024
E.g.f.: exp(x)*(1 + 18*x + 48*x^2/2! + 32*x^3/3!).Note that -T(6, i*sqrt(x)) = 1 + 18*x + 48*x^2 + 32*x^3, where T(n, x) denotes the Chebyshev polynomial of the first kind. SeeA008310.(End)