cp's OEIS Frontend

A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).

%I A113319 #49 Apr 28 2025 07:43:34
%S A113319 2,0,7,6,6,7,4,0,4,7,4,6,8,5,8,1,1,7,4,1,3,4,0,5,0,7,9,4,7,5,0,0,0,0,
%T A113319 4,9,0,4,4,5,6,5,6,2,6,6,4,0,3,8,1,6,6,6,5,5,7,5,0,6,2,4,8,4,3,9,0,1,
%U A113319 5,4,2,4,7,9,1,8,3,1,0,0,2,1,7,4,3,5,6,5,5,5,1,7,5,9,3,9,5,4,9,1,8,7,6,5,1
%N A113319 Decimal expansion of Sum_{k>=0} 1/(k^2+1).
%C A113319 Known to be transcendental. After n=2 it is the same as A100554(n).
%C A113319 Imaginary part of psi(I) (for the real part, see A248177). - _Stanislav Sykora_, Oct 03 2014
%D A113319 Michel Waldschmidt, Elliptic functions and transcendance, Surveys in number theory, 143-188, Dev. Math., 17, Springer, New York, 2008.
%H A113319 Ivan Panchenko, <a href="/A113319/b113319.txt">Table of n, a(n) for n = 1..1000</a>
%H A113319 Wikipedia, <a href="http://en.wikipedia.org/wiki/Digamma_function">Digamma function</a>.
%H A113319 <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.
%F A113319 Equals 1/2 + Pi /(2*tanh(Pi)).
%F A113319 Equals 1+Integral_{x >= 0} sin(x)/(exp(x)-1) dx. - _Robert FERREOL_, Jan 12 2016.
%F A113319 Equals Sum_{k>=0} (-1)^(k+1)*(zeta(2*k) - 1). - _Amiram Eldar_, Apr 28 2025
%e A113319 2.076674047468581174134050794750000490445656266403816665575062484390...
%t A113319 RealDigits[N[Im[PolyGamma[0,I]],105]][[1]]  (* _Vaclav Kotesovec_, Oct 03 2014 *)
%o A113319 (PARI) 1/2+Pi/tanh(Pi)/2
%o A113319 (PARI) imag(psi(I)) \\ _Stanislav Sykora_, Oct 03 2014
%o A113319 (PARI) sumnumrat(1/(x^2+1),0) \\ _Charles R Greathouse IV_, Jan 20 2022
%Y A113319 Cf. A013661 (Sum_{i>=1} 1/i^2), A232883 (Sum_{i>=0} 1/(2*i^2+1)). - _Bruno Berselli_, Dec 02 2013
%Y A113319 Cf. A248177.
%Y A113319 Essentially the same as A100554 and A259171.
%K A113319 nonn,cons
%O A113319 1,1
%A A113319 _Benoit Cloitre_, Jan 07 2006
%E A113319 Offset changed from 0 to 1 by _Bruno Berselli_, Dec 02 2013