%I #14 Sep 08 2022 08:46:20

%S 1,11,83,410,16799,495151,8516747,55850623,309309419,1088610631,

%T 6561497681,777210281963,12475578306953,287734917200239,

%U 10671842976127147,844855135994501953,846430303832665873,75457260356268267017,3551759427031132995079,711302288219532928235,712163917143684270659

%N Numerators of the partial sums of the reciprocals of the numbers (k + 1)*(5*k + 4) = 2*A005476(k+1), for k >= 0.

%C The corresponding denominators are given in A294832.

%C 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].

%C 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.

%C 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)).

%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189-193. For v_5(4) see p. 192.

%H G. C. Greubel, <a href="/A294831/b294831.txt">Table of n, a(n) for n = 0..600</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>

%F 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.

%e 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, ...

%e V(5,4;10^6) = 0.3877927018 (Maple 10 digits) to be compared with 0.3877929018 obtained from A294833 with 10 digits.

%t Table[Numerator[Sum[1/((k + 1)*(5*k + 4)), {k, 0, n}]], {n, 0, 50}] (* _G. C. Greubel_, Aug 30 2018 *)

%o (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 4)))); \\ _Michel Marcus_, Nov 19 2017

%o (Magma) [Numerator((&+[1/((k + 1)*(5*k + 4)): k in [0..n]])): n in [0..50]]; // _G. C. Greubel_, Aug 30 2018

%Y Cf. A001622, 2*A005476, A294512, A294828/A294829 (V(5,3;n)), A294832, A294833.

%K nonn,frac,easy

%O 0,2

%A _Wolfdieter Lang_, Nov 18 2017