A196521 Decimal expansion of Pi/4-log(2)/2.

4, 3, 8, 8, 2, 4, 5, 7, 3, 1, 1, 7, 4, 7, 5, 6, 5, 4, 9, 0, 7, 0, 4, 4, 7, 8, 5, 0, 9, 0, 7, 8, 7, 4, 3, 7, 0, 1, 1, 5, 4, 2, 2, 8, 2, 6, 6, 3, 6, 4, 8, 8, 2, 8, 1, 8, 3, 3, 9, 6, 1, 4, 3, 3, 3, 0, 2, 5, 7, 2, 9, 0, 5, 8, 6

Offset

0,1

References

L. B. W. Jolley, Summation of series, Dover Publications Inc., New York, 1961, p. 14 (eq. 72).

Links

Table of n, a(n) for n=0..74.

Jean-Paul Allouche and Jeffrey Shallit, Sums of digits and the Hurwitz zeta function, in: K. Nagasaka and E. Fouvry (eds.), Analytic Number Theory, Lecture Notes in Mathematics, Vol. 1434, Springer, Berlin, Heidelberg, 1990, pp. 19-30.

Formula

Equals 1 - 1/2 - 1/3 + 1/4 + 1/5 - ....

Equals Sum_{n>=0} 2/((4*n+2)*(4*n+3)). - Peter Luschny, Dec 06 2013

Equals Sum_{n>=1} (-1)^(n+1)/((2*n-1)*(2*n)). - Robert FERREOL, Dec 14 2015

Equals Integral_{x=0..1} (arctan(x)) dx = Integral_{x=0..Pi/4} (x / cos(x)^2) dx = Integral_{x=0..1/sqrt(2)} (arcsin(x)/(1-x^2)^(3/2)) dx. - Robert FERREOL, Dec 14 2015

Equals Integral_{x>=0} (exp(x) - 1)/(exp(2*x) + 1) dx. - Peter Bala, Nov 01 2019

From Bernard Schott, Sep 07 2020: (Start)

Equals Sum_{n>=1} (-1)^(n*(n-1)/2) / n [compare with A231902 formula].

Equals Sum_{n>=0} (8*n+5) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)

Equals Sum_{k>=1} A033264(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021

Example

0.438824573117475654907044785090787437011542282663648828183396143330257...

Maple

Digits:=100; evalf(Pi/4-log(2)/2); # Wesley Ivan Hurt, Dec 06 2013

Mathematica

RealDigits[Pi/4 - Log[2]/2, 10, 100] (* Wesley Ivan Hurt, Dec 06 2013 *)

Prog

(PARI) Pi/4-log(2)/2 \\ Altug Alkan, Dec 14 2015

Crossrefs

Cf. A003881, A016655 (10*log(2)/2)), A033264.

Cf. A231902 (Pi/4+log(2)/2).

Sequence in context: A089368 A357130 A116583 * A021699 A131416 A134390

Adjacent sequences: A196518 A196519 A196520 * A196522 A196523 A196524

Keyword

nonn,cons,easy

Author

R. J. Mathar, Oct 03 2011

Status

approved