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 A294520 #17 Feb 16 2025 08:33:51
%S A294520 1,13,49,795,84179,366829,11417459,103067441,4235695001,97604192047,
%T A294520 1661825059679,1663957022369,101611584435869,101706166053389,
%U A294520 7226964017429851,17176158550059533,154681745346189277,6654999228519884521,6658297729691103841,21316057915886595965,2153790894613123442641
%N A294520 Numerators of the partial sums of the reciprocals of the dodecagonal numbers (k + 1)*(5*k + 1) = A051624(k+1), for k >= 0.
%C A294520 The corresponding denominators are given in A294521.
%C A294520 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] = [5,1].
%C A294520 The limit of the series is V(5,1) = lim_{n -> oo} V(5,1;n) = ((5/2)*log(5) + (2*phi - 1)*(log(phi) + (Pi/5)*sqrt(3 + 4*phi)))/8, with the golden section phi:= (1 + sqrt(5))/2. The value is 1.17795605792266... given in A244649.
%D A294520 Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, Eulersche Reihen, pp. 189 - 193.
%H A294520 G. C. Greubel, <a href="/A294520/b294520.txt">Table of n, a(n) for n = 0..600</a>
%H A294520 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/DigammaFunction.html">Digamma Function</a>
%F A294520 a(n) = numerator(V(5,1;n)) with V(5,1;n) = Sum_{k=0..n} 1/((k + 1)*(5*k + 1)) = Sum_{k=0..n} 1/A051624(k+1) = (1/4)*Sum_{k=0..n} (1/(k + 1/5) - 1/(k+1)) = (-Psi(1/5) + Psi(n+6/5) - (gamma + Psi(n+2)))/4, with the digamma function Psi and the Euler-Mascheroni constant gamma = -Psi(1) from A001620.
%e A294520 The rationals V(5,1;n), n >= 0, begin: 1, 13/12, 49/44, 795/704, 84179/73920, 366829/320320, 11417459/9929920, 103067441/89369280, 4235695001/3664140480, 97604192047/84275231040, 1661825059679/1432678927680, ...
%e A294520 V(5,1;10^6) = 1.177956058 (Maple, 10 digits) to be compared with 1.177956058 obtained from V(5,1) given in A244649.
%t A294520 Table[Numerator[Sum[1/((k + 1)*(5*k + 1)), {k, 0, n}]], {n, 0, 30}] (* _G. C. Greubel_, Aug 29 2018 *)
%o A294520 (PARI) a(n) = numerator(sum(k=0, n, 1/((k + 1)*(5*k + 1)))); \\ _Michel Marcus_, Nov 15 2017
%o A294520 (Magma) [Numerator((&+[1/((k+1)*(5*k+1)): k in [0..n]])): n in [0..25]]; // _G. C. Greubel_, Aug 29 2018
%Y A294520 Cf. A001620, A051624, A244649, A294512, A294516/A294517, A294521.
%K A294520 nonn,frac,easy
%O A294520 0,2
%A A294520 _Wolfdieter Lang_, Nov 15 2017