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A294516
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Numerators of the partial sums of the reciprocals of (k+1)*(4*k+3) = A033991(k+1), for k >= 0.
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3
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1, 17, 67, 2087, 40577, 315967, 8627249, 539432053, 543008461, 7096662277, 306487877071, 14457409539227, 246534893826499, 49437672710843, 14617658229054773, 29294219493288391, 1966205309547985477, 139821581165897995307, 700098935135639210887, 55378426713778630607653, 4601722042202662057443599, 12144567347216934480292961
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OFFSET
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0,2
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COMMENTS
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The corresponding numerators are given in A294517.
For the general case V(m,r;n) = Sum_{k=0..n} 1/((k + 1)*(m*k + r)) = (1/(m - r))*Sum_{k=0..n} (m/(m*k + r) - 1/(k+1)), for r = 1, ..., m-1 and m = 2, 3, ..., and their limits see a comment in A294512. Here [m,r] = [4,3].
The limit of the series is V(4,3) = lim_{n -> oo} V(4,3;n) = 3*log(2) - Pi/2 = 0.50864521488493930902... given in A294518.
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REFERENCES
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Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.
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LINKS
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FORMULA
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a(n) = numerator(V(4,3;n)) with V(4,3;n) = Sum_{k=0..n} 1/((k + 1)*(4*k + 3)) = Sum_{k=0..n} 1/A033991(k+1) = Sum_{k=0..n} (4/(4*k + 3) - 1/(k+1)).
V(4,3;n) = 3*log(2) - Pi/2 + Psi(n+7/4) - Psi(n+2) with the digamma function Psi. Note that Psi(1) - Psi(3/4) = 3*log(2) - Pi/2. - Wolfdieter Lang, Nov 15 2017
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EXAMPLE
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The rationals V(4,3;n), n >= 0, begin: 1/3, 17/42, 67/154, 2087/4620, 40577/87780, 315967/672980, 8627249/18170460, 539432053/1126568520, 543008461/1126568520, 7096662277/14645390760, 306487877071/629751802680, ...
V(4,3;10^4) = 0.508620219 (Maple, 10 digits) to be compared with 0.508645215 from V(4,3) given in A294518.
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PROG
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(PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(4*k + 3)))); \\ Michel Marcus, Nov 15 2017
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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