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 A381028 #24 Feb 16 2025 01:18:37
%S A381028 4,3,7,6,5,8,2,4,2,3,1,1,2,6,1,0,9,3,3,1,5,9,2,0,9,2,6,4,3,8,0,5,1,4,
%T A381028 0,1,6,4,8,4,3,5,6,4,5,3,5,2,3,0,6,9,6,8,3,0,2,7,1,5,6,1,3,1,5,1,3,3,
%U A381028 2,3,4,3,5,7,1,5,8,9,4,1,7,2,4,1,6,0,1,6,8,3,9,4,9,8,3,0,9,8,5,4,2,3,9,3,1
%N A381028 Decimal expansion of Sum_{k>=1} zeta(2k)/((2k-1)*2^(2k)).
%C A381028 Including the term k=0 with zeta(0) = -1/2 gives 0.937658... = this + 1/2.
%H A381028 H. M. Srivastava, M. L. Glasser, and V. S. Adamchik, <a href="https://doi.org/10.4171/ZAA/982">Some definite integrals associated with the Riemann Zeta Function</a>, Z. Anal. Anw. 19 (3) (2000) 831-846, (2.18) at n=0.
%F A381028 4*this = 1.75063296924504437... = Sum_{k>=1} (1/k)*log((2k+1)/(2k-1)).
%F A381028 Equals Sum_{k>=1} arctanh(1/(2*k))/(2*k) = Sum_{k>=1} arccoth(2*k)/(2*k). - _Amiram Eldar_, Feb 15 2025
%e A381028 0.4376582423112610933159209...
%p A381028 evalf(Sum(Zeta(2*k)/((2*k-1)*4^k), k = 1 .. infinity), 105) # _Amiram Eldar_, Feb 15 2025
%t A381028 RealDigits[NSum[Zeta[2*k]/((2*k - 1)*4^k), {k, 1, Infinity}, WorkingPrecision -> 120, NSumTerms -> 200]][[1, 1 ;; 105]] (* _Amiram Eldar_, Feb 15 2025 *)
%o A381028 (PARI) sumpos(k=1, zeta(2*k)/((2*k-1)*2^(2*k))) \\ _Michel Marcus_, Feb 13 2025
%Y A381028 Cf. A256318, A355922.
%K A381028 nonn,cons
%O A381028 0,1
%A A381028 _R. J. Mathar_, Feb 12 2025
%E A381028 More terms from _Amiram Eldar_, Feb 15 2025