%I #78 Nov 05 2025 15:22:21

%S 6,2,3,2,2,5,2,4,0,1,4,0,2,3,0,5,1,3,3,9,4,0,2,0,0,8,0,2,5,0,5,6,8,0,

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

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

%N Decimal expansion of log(1+sqrt(2))/sqrt(2).

%H G. C. Greubel, <a href="/A196525/b196525.txt">Table of n, a(n) for n = 0..10000</a>

%H R. J. Mathar, <a href="https://arxiv.org/abs/1008.2547">Table of Dirichlet L-series and Prime Zeta Modulo Functions for Small Moduli</a>, arXiv:1008.2547 [math.NT], 2010-2015, Table section 2.2, L(m=8, r=2, s=1).

%H Paul J. Nahin, <a href="https://doi.org/10.1007/978-3-030-43788-6">Inside interesting integrals</a>, Undergrad. Lecture Notes in Physics, Springer (2020), (2.2.3)

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>.

%F Equals Sum_{n>=1} A091337(n)/n = 1 - 1/3 - 1/5 + 1/7 + 1/9 - 1/11 - ...

%F Equals 2*Sum_{n>=1} (-1)^n/A001539(n). - _Michel Marcus_, Sep 27 2017

%F From _Fred Daniel Kline_, May 23 2019: (Start)

%F Equals arcsinh(1)/sqrt(2).

%F Equals Sum_{n>=1} 1/A118417(n-1) = Sum_{n>=1} 1/((2*n - 1)*2^n). (End)

%F From _Peter Bala_, Nov 01 2019: (Start)

%F Equals (1/sqrt(2))*arccoth(sqrt(2)).

%F Equals 1 - 8*Sum_{n >= 0} (-1)^(n+1)*n/(16*n^2 - 1).

%F Equals 1 - Integral_{x = 0..inf} exp(-2*x)*cosh(x)/cosh(2*x) dx.

%F Equals 2*Integral_{x = 0..inf} exp(x)*(exp(2*x) + 1)*(exp(4*x) - 1)/(exp(4*x) + 1)^2 dx - 1. (End)

%F From _Amiram Eldar_, Aug 16 2020: (Start)

%F Equals Sum_{k>=0} (-1)^k * (2*k)!!/(2*k+1)!!.

%F Equals Integral_{x=0..Pi/4} 1/(cos(x) + sin(x)) dx. (End)

%F From _Peter Bala_, Dec 01 2021: (Start)

%F Equals 2*Sum_{k >= 0} (-1)^k/((4*k + 1)*(4*k + 3)).

%F Let N be a positive integer divisible by 4. We have the asymptotic expansion (1/sqrt(2))*log(1 + sqrt(2)) - 2*Sum_{k = 0..N/4 - 1} (-1)^k/((4*k + 1)*(4*k + 3)) ~ 1/N^2 - 11/N^4 + 361/N^6 - 24611/N^8 + ..., where the sequence of unsigned coefficients [1, 11, 361, 24611, ...] is A000464. See A181048 and A181049. An example is given below. (End)

%F Equals 1/Product_{p prime} (1 - Kronecker(8,p)/p), where Kronecker(8,p) = 0 if p = 2, 1 if p == 1 or 7 (mod 8) or -1 if p == 3 or 5 (mod 8). - _Amiram Eldar_, Dec 17 2023

%F Equals integral_{x=0..Pi/2} sin^2(x)/(sin(x)+cos(x)) dx [Nahin]. - _R. J. Mathar_, May 16 2024

%e 0.6232252401402305133940200802505680... = A091648/A002193.

%e From _Peter Bala_, Dec 01 2021: (Start)

%e With N = 10000, the truncated series Sum_{k = 0..N/4 - 1} (-1)^k/((4*k + 1)*(4*k+3)) = 0.6232252[3]014023[16]1339[3659]080... to 27 decimal places. The square bracketed numbers show where this decimal expansion differs from that of (1/sqrt(2))*log(1+sqrt(2)) = 0.6232252(4)014023(05) 1339(4020)080.... The numbers 1, -11, 361 must be added to the square bracketed numbers to give the correct decimal expansion to 27 decimal places. (End)

%t RealDigits[Log[1+Sqrt[2]]/Sqrt[2],10,120][[1]] (* _Harvey P. Dale_, Dec 27 2011 *)

%t RealDigits[Sum[1/((2 n - 1) 2^n), {n, 1, Infinity}], 10, 120][[1]] (* _Fred Daniel Kline_, May 23 2019 *)

%o (PARI) log(sqrt(2)+1)/sqrt(2) \\ _Michel Marcus_, Sep 27 2017

%o (Magma) SetDefaultRealField(RealField(100)); Log(Sqrt(2)+1)/Sqrt(2); // _G. C. Greubel_, Oct 05 2018

%Y Cf. A000464, A002193, A001539, A091648, A181048, A181049.

%K nonn,cons,easy

%O 0,1

%A _R. J. Mathar_, Oct 03 2011