A231902 Decimal expansion of Pi/4 + log(2)/2.

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

Offset: 1

Author

Bruno Berselli, Nov 15 2013

Keywords

• nonn
• cons

Examples

			1.131971753677420964324276906548964005087042417023904082304076152823650...
		

References

• L. B. W. Jolley, Summation of series, Dover Publications Inc. (New York), 1961, p. 28 (formula 154).
• Jean-Marie Monier, Analyse, Exercices corrigés, 2ème année MP, Dunod, 1997, Exercice 3.15, p. 269.

Links

• Ivan Panchenko, Table of n, a(n) for n = 1..1000
• 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.
• Michael Penn, A nice integral, YouTube video, 2022.

Crossrefs

Cf. A003881 (Pi/4), A016655 (10*(log(2)/2)), A072691 (Pi^2/12).

Cf. A006752 (Catalan's constant)

Cf. A196521 (Pi/4-log(2)/2).

Cf. A037800.

Programs

Magma

SetDefaultRealField(RealField(100)); R:=RealField(); (Pi(R) + 2*Log(2))/4; // G. C. Greubel, Aug 24 2018

Mathematica

RealDigits[Pi/4 + Log[2]/2, 10, 90][[1]]

PARI

default(realprecision, 100); (Pi + 2*log(2))/4 \\ G. C. Greubel, Aug 24 2018

Formula

Equals 1 + Sum_{m>=1} -(-1)^m/(2*m*(2*m+1)) = 1 + 1/(2*3) - 1/(4*5) + 1/(6*7) - 1/(8*9) + ... .

From Amiram Eldar, Jul 16 2020: (Start)

Equals Integral_{x=1..oo} arctan(x)/x^2 dx.

Equals 1 + Integral_{x=0..1/2} log(4*x^2 + 1) dx. (End)

From Bernard Schott, Sep 07 2020: (Start)

Equals -Sum_{n>=1} (-1)^(n*(n+1)/2) / n [compare with A196521 formula].

Equals Sum_{n>=0} (32*n^2+40*n+11) / (4*(n+1)*(2*n+1)*(4*n+1)*(4*n+3)). (End)

Equals 1 + Sum_{k>=1} A037800(k)/(k*(k+1)) (Allouche and Shallit, 1990). - Amiram Eldar, Jun 01 2021