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A294831
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Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 4) = 2*A005476(k+1), for k >= 0.
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3
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1, 11, 83, 410, 16799, 495151, 8516747, 55850623, 309309419, 1088610631, 6561497681, 777210281963, 12475578306953, 287734917200239, 10671842976127147, 844855135994501953, 846430303832665873, 75457260356268267017, 3551759427031132995079, 711302288219532928235, 712163917143684270659
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OFFSET
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0,2
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COMMENTS
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The corresponding denominators are given in A294832.
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] = [5,4].
The limit of the series is V(5,4) = lim_{n -> oo} V(5,4;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) - (Pi/5)*sqrt(3 + 4*phi)))/2, with the golden section phi:= (1 + sqrt(5))/2 = A001622. The value is 0.3877929018046... given in A294833.
In the Koecher reference v_5(4) = (1/5)*V(5,4) = 0.07755858036... given there by (1/4)*log(5) + (1/(2*sqrt(5)))*log((1 + sqrt(5))/2) - (Pi/10)*sqrt((5 + 2*sqrt(5))/5)).
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REFERENCES
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Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.
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LINKS
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FORMULA
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a(n) = numerator(V(5,4;n)) with V(5,4;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 4)) = Sum_{k=0..n} 1/(2*A005476(k+1)) = Sum_{k=0..n} (1/(k + 4/5) - 1/(k+1)) = -Psi(4/5) + Psi(n+9/5) - (gamma + Psi(n+2)) with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.
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EXAMPLE
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The rationals V(5,4;n), n >= 0, begin:1/4, 11/36, 83/252, 410/1197, 16799/47880, 495151/1388520, 8516747/23604840, 55850623/153431460, 309309419/843873030, 1088610631/2953555605, 6561497681/17721333630, 777210281963/2091117368340, 12475578306953/33457877893440, ...
V(5,4;10^6) = 0.3877927018 (Maple 10 digits) to be compared with 0.3877929018 obtained from A294833 with 10 digits.
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MATHEMATICA
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Table[Numerator[Sum[1/((k + 1)*(5*k + 4)), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Aug 30 2018 *)
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PROG
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(PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 4)))); \\ Michel Marcus, Nov 19 2017
(Magma) [Numerator((&+[1/((k + 1)*(5*k + 4)): k in [0..n]])): n in [0..50]]; // G. C. Greubel, Aug 30 2018
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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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