This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A294516 #14 Feb 16 2025 08:33:51
%S A294516 1,17,67,2087,40577,315967,8627249,539432053,543008461,7096662277,
%T A294516 306487877071,14457409539227,246534893826499,49437672710843,
%U A294516 14617658229054773,29294219493288391,1966205309547985477,139821581165897995307,700098935135639210887,55378426713778630607653,4601722042202662057443599,12144567347216934480292961
%N A294516 Numerators of the partial sums of the reciprocals of (k+1)*(4*k+3) = A033991(k+1), for k >= 0.
%C A294516 The corresponding numerators are given in A294517.
%C A294516 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].
%C A294516 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.
%D A294516 Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193.
%H A294516 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>
%F A294516 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)).
%F A294516 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
%e A294516 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, ...
%e A294516 V(4,3;10^4) = 0.508620219 (Maple, 10 digits) to be compared with 0.508645215 from V(4,3) given in A294518.
%o A294516 (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(4*k + 3)))); \\ _Michel Marcus_, Nov 15 2017
%Y A294516 Cf. A294512, A250551(n+1)/A294515(n) (V(4,1;n)), A294517, A294518.
%K A294516 nonn,frac,easy
%O A294516 0,2
%A A294516 _Wolfdieter Lang_, Nov 07 2017