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 A275792 #35 Mar 25 2024 09:59:10
%S A275792 1,1,5,0,9,8,2,3,6,8,0,9,4,6,7,6,3,8,6,3,6,3,6,8,9,8,9,6,9,5,2,6,7,5,
%T A275792 0,5,8,3,0,9,6,6,7,0,9,5,5,1,8,7,4,9,1,0,9,8,3,9,6,4,5,7,8,4,5,0,5,0,
%U A275792 4,2,6,9,1,0,9,1,3,6,6,7,4,1,4,0,9,6,6,7,5,5,3,7,0,6,3,0,5,1,5
%N A275792 Decimal expansion of the sum of the reciprocals of the tetradecagonal numbers A051866.
%C A275792 See Table 1 of the Downey et al. link.
%C A275792 From _Wolfdieter Lang_, Nov 09 2017: (Start)
%C A275792 The general formula for S_{2*(k+1)} = Sum_{n>=0} 1/((n+1)*(k*n+1)) given in the Downey et al. link is a special case of the simpler formula for V(m,r) = Sum_{n>=0} 1/((n+1)*(m*n + r)), r = 1,2, ... ,m -1. V(m,r) = (m/(m-r))*v_m(r) in Koecher's notation. For this formula for m*v_m(r) see a comment in A294512.
%C A275792 The special case is m = k and r = 1, leading to S_{2*(k+1)} = V(k,1) = (log(k) + (Pi/2)*cot(Pi/k) - Sum_{j=1..k-1} cos(2*Pi*j/k)*log(2*sin(Pi*j/k)))/(k-1), for k >= 2.
%C A275792 S_14, for k=6, is then given by the formula below (also obtained from the more complicated formula of Downey et al.).
%C A275792 The partial sums are given in A294834/A294835.
%C A275792 (End)
%D A275792 Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193. See (6/5)*v_6(1) on p. 192.
%H A275792 G. C. Greubel, <a href="/A275792/b275792.txt">Table of n, a(n) for n = 1..10000</a>
%H A275792 Lawrence Downey, Boon W. Ong, and James A. Sellers, <a href="https://www.d.umn.edu/~jsellers/downey_ong_sellers_cmj_preprint.pdf">Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers</a>, Coll. Math. J., 39, no. 5 (2008), 391-394.
%F A275792 Sum_{n >= 1} 1/(n*(6*n - 5)) = 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10.
%e A275792 1.150982368094676386363689896952675058309...
%t A275792 RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* _Amiram Eldar_, Jun 25 2023 *)
%o A275792 (PARI) 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ _Michel Marcus_, Nov 09 2017
%o A275792 (Magma) SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // _G. C. Greubel_, Mar 25 2024
%o A275792 (SageMath) numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # _G. C. Greubel_, Mar 25 2024
%Y A275792 Cf. A013661, A016627, A051866, A244641, A244645, A294512, A294834, A294835.
%K A275792 nonn,cons,easy
%O A275792 1,3
%A A275792 _Wolfdieter Lang_, Sep 12 2016