|
%I #32 Apr 09 2022 06:15:33
%S 1,4,4,0,6,5,9,5,1,9,9,7,7,5,1,4,5,9,2,6,5,8,9,3,2,5,0,2,9,1,3,9,8,1,
%T 7,1,2,5,2,5,2,9,7,0,8,4,6,7,3,6,5,8,6,9,0,8,2,1,6,1,4,0,9,2,4,6,2,0,
%U 3,1,1,5,2,2,3,3,5,6,6,0,7,7,6,4,7,9
%N Decimal expansion of (Pi/2)*tanh(Pi/2).
%C The old name was: Decimal expansion of sum of reciprocals, main diagonal of the natural number array, A185787.
%C Let s(n) be the sum of reciprocals of the numbers in row n of the array T at A185787 given by T(n,k) = n + (n+k-2)(n+k-1)/2, and let r = (2*pi/sqrt(7))*tanh(pi*sqrt(7)/2), as at A226985. Then s(1) = r, and s(2) to s(5) are given by A228044 to A228047.
%C Let c(n) be the sum of reciprocals of the numbers in column n of T. Then c(1) = 2; c(2) = 11/9, c(4) = 29/50, and c(3) is given by A228049. Let d(n) be the sum of reciprocals of the numbers in the main diagonal, (T(n,n)); then d(2) = (1/12)*(pi)^2; d(3) = 1/2, and d(1) is given by A228048.
%C This is also the value of the series 1 + 2*Sum_{n>=1} 1/(4*n^4 + 1) = 1 + 2*(1/5 + 1/65 + 1/325 + ...). See the Koecher reference, p. 189. - _Wolfdieter Lang_, Oct 30 2017
%D Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, p. 189.
%F Equals Sum_{k>=0} 1/A001844(k). - _Amiram Eldar_, Jun 20 2020
%F Equals Integral_{x=0..oo} sin(x)/sinh(x) dx. - _Amiram Eldar_, Aug 10 2020
%F Equals Product_{k>=2} ((k^2 + 1)/(k^2 - 1))^((-1)^k). - _Amiram Eldar_, Apr 09 2022
%e 1/1 + 1/5 + 1/13 + ... = (Pi/2)*tanh(Pi/2) = 1.4406595199775145926589...
%t $MaxExtraPrecision = Infinity; t[n_, k_] := t[n, k] = n + (n + k - 2) (n + k - 1)/2; u = N[Sum[1/t[n, n], {n, 1, Infinity}], 130]; RealDigits[u, 10]
%t RealDigits[Pi*Tanh[Pi/2]/2, 10, 100][[1]] (* _Amiram Eldar_, Apr 09 2022 *)
%o (PARI) (Pi/2)*tanh(Pi/2) \\ _Michel Marcus_, Jun 20 2020
%Y Cf. A000027, A001844, A185787, A226985, A228044.
%K nonn,cons,easy
%O 1,2
%A _Clark Kimberling_, Aug 06 2013
%E Name changed by _Wolfdieter Lang_, Oct 30 2017
|
