A228048 Decimal expansion of (Pi/2)*tanh(Pi/2).

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, 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, 3, 1, 1, 5, 2, 2, 3, 3, 5, 6, 6, 0, 7, 7, 6, 4, 7, 9

Offset: 1

Author

Clark Kimberling, Aug 06 2013

Keywords

• nonn
• cons
• easy

Comments

The old name was: Decimal expansion of sum of reciprocals, main diagonal of the natural number array, A185787.

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.

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.

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

Examples

			1/1 + 1/5 + 1/13 + ... = (Pi/2)*tanh(Pi/2) = 1.4406595199775145926589...
		

References

• Max Koecher, Klassische elementare Analysis, Birkhäuser, Basel, Boston, 1987, p. 189.

Crossrefs

Cf. A000027, A001844, A185787, A226985, A228044.

Programs

Mathematica

$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]
RealDigits[Pi*Tanh[Pi/2]/2, 10, 100][[1]] (* Amiram Eldar, Apr 09 2022 *)

PARI

(Pi/2)*tanh(Pi/2) \\ Michel Marcus, Jun 20 2020

Formula

Equals Sum_{k>=0} 1/A001844(k). - Amiram Eldar, Jun 20 2020

Equals Integral_{x=0..oo} sin(x)/sinh(x) dx. - Amiram Eldar, Aug 10 2020

Equals Product_{k>=2} ((k^2 + 1)/(k^2 - 1))^((-1)^k). - Amiram Eldar, Apr 09 2022

Extensions

Name changed by Wolfdieter Lang, Oct 30 2017