

A228048


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


9



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
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OFFSET

1,2


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+k2)(n+k1)/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


REFERENCES

Max Koecher, Klassische elementare Analysis, BirkhĂ¤user, Basel, Boston, 1987, p. 189.


LINKS

Table of n, a(n) for n=1..86.


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


EXAMPLE

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


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]


PROG

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


CROSSREFS

Cf. A000027, A001844, A185787, A226985, A228044.
Sequence in context: A246811 A197140 A155502 * A016705 A245592 A169783
Adjacent sequences: A228045 A228046 A228047 * A228049 A228050 A228051


KEYWORD

nonn,cons,easy


AUTHOR

Clark Kimberling, Aug 06 2013


EXTENSIONS

Name changed by Wolfdieter Lang, Oct 30 2017


STATUS

approved



