OFFSET
1,3
COMMENTS
See Table 1 of the Downey et al. link.
From Wolfdieter Lang, Nov 09 2017: (Start)
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.
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.
S_14, for k=6, is then given by the formula below (also obtained from the more complicated formula of Downey et al.).
(End)
REFERENCES
Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, pp. 189 - 193. See (6/5)*v_6(1) on p. 192.
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..10000
Lawrence Downey, Boon W. Ong, and James A. Sellers, Beyond the Basel Problem: Sums of Reciprocals of Figurate Numbers, Coll. Math. J., 39, no. 5 (2008), 391-394.
FORMULA
Sum_{n >= 1} 1/(n*(6*n - 5)) = 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10.
EXAMPLE
1.150982368094676386363689896952675058309...
MATHEMATICA
RealDigits[2*Log[2]/5 + 3*Log[3]/10 + Sqrt[3]*Pi/10, 10, 120][[1]] (* Amiram Eldar, Jun 25 2023 *)
PROG
(PARI) 2*log(2)/5 + 3*log(3)/10 + sqrt(3)*Pi/10 \\ Michel Marcus, Nov 09 2017
(Magma) SetDefaultRealField(RealField(139)); R:= RealField(); (4*Log(2) + 3*Log(3) + Pi(R)*Sqrt(3))/10; // G. C. Greubel, Mar 25 2024
(SageMath) numerical_approx((4*log(2) + 3*log(3) + pi*sqrt(3))/10, digits=139) # G. C. Greubel, Mar 25 2024
CROSSREFS
KEYWORD
AUTHOR
Wolfdieter Lang, Sep 12 2016
STATUS
approved