cp's OEIS Frontend

A345203 Decimal expansion of zeta(2) + 2*zeta(3).

%I A345203 #6 Jun 10 2021 22:36:25
%S A345203 4,0,4,9,0,4,7,8,7,3,1,6,7,4,1,5,0,0,7,2,7,1,8,9,1,4,8,9,6,6,8,9,2,5,
%T A345203 1,7,0,7,4,8,9,2,2,4,8,5,8,8,7,7,9,6,2,0,1,3,2,0,1,0,1,3,4,0,0,5,3,6,
%U A345203 8,3,8,8,1,9,7,5,8,2,7,0,5,4,2,0,6,5,4
%N A345203 Decimal expansion of zeta(2) + 2*zeta(3).
%C A345203 Ovidiu Furdui, Limits, Series, and Fractional Part Integrals, Springer, 2013, section 3.71, p. 150.
%H A345203 Ovidiu Furdui, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.371">Series Involving Products of Two Harmonic Numbers</a>, Mathematics Magazine, Vol. 84, No. 5 (2011), pp. 371-377.
%F A345203 Equals A013661 + 2 * A002117.
%F A345203 Equals Sum_{k>=1} (k+2)/k^3.
%F A345203 Equals Sum_{k>=1} H(k)*H(k+1)/(k*(k+1)), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number (Furdui, 2011).
%F A345203 Equals Sum_{k>=1} (H(k)+1)/k^2.
%F A345203 Equals 1 + Sum_{k>=2} H(k)/(k-1)^2.
%F A345203 Equals Sum_{k>=2} (k-1)^2*(zeta(k)-1).
%F A345203 Equals 3 + Sum_{k>=3} (-1)^(k+1)*k^2*(zeta(k)-1).
%F A345203 Equals Integral_{x=0..1} log(x)*(log(x)-1)/(1-x) dx.
%F A345203 Equals Integral_{x>=1} log(x)*(log(x)+1)/(x*(x-1)) dx.
%F A345203 Equals Integral_{x>=0} x*(x+1)/(exp(x)-1) dx.
%e A345203 4.04904787316741500727189148966892517074892248588779...
%t A345203 RealDigits[Zeta[2] + 2*Zeta[3], 10, 100][[1]]
%Y A345203 Cf. A001008, A002117, A002805, A013661, A218505, A345204.
%K A345203 nonn,cons
%O A345203 1,1
%A A345203 _Amiram Eldar_, Jun 10 2021