A002162 -id:A002162 - cp's OEIS frontend

A152683 Decimal expansion of log_6 (2).

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

The upper bound for the ratio of the number of 3x+1 steps to all steps in the Collatz iteration. - T. D. Noe, Apr 30 2010

			.38685280723454158687024613846782087646514185945710342838949...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Emmanuel Jeandel, Michael Rao, An aperiodic set of 11 Wang tiles, arXiv:1506.06492 [cs.DM], 2015. See p. 10.
Index entries for transcendental numbers

Cf. decimal expansion of log_6(m): this sequence, A152935 (m=3), A153102 (m=4), A153202 (m=5), A153617 (m=7), A153754 (m=8), A154009 (m=9), A154157 (m=10), A154178 (m=11), A154199 (m=12), A154278 (m=13), A154466 (m=14), A154567 (m=15), A154776 (m=16), A154856 (m=17), A154911 (m=18), A155044 (m=19), A155490 (m=20), A155554 (m=21), A155697 (m=22), A155823 (m=23), A155959 (m=24).

SetDefaultRealField(RealField(100)); Log(2)/Log(6); // G. C. Greubel, Sep 13 2018

RealDigits[Log[6,2],10,120][[1]] (* Harvey P. Dale, Sep 12 2012 *)

default(realprecision, 100); log(2)/log(6) \\ G. C. Greubel, Sep 13 2018

Equals log(2)/log(6) (A002162/A016629), that is, log(2)/(log(2)+log(3)). - Michel Marcus, Aug 18 2018

A061444 Decimal expansion of log(2 * Pi).

Original entry on oeis.org

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

Offset: 1

Frank Ellermann, Jun 11 2001

Used in formulas for gamma(x), e.g., in Stirling's approximation for m!.
Also decimal expansion of zeta'(0)/zeta(0). - Benoit Cloitre, Sep 28 2002
The value of log(2*Pi) is close to 1 + Sum_{n>=2} log(zeta(n)) = 1.83067035427178011248.... - Arkadiusz Wesolowski, Jul 17 2011

			1.837877066409345483560659472811235279722794947275566825634303...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Simon Plouffe, log(2*Pi) to 10000 digits
Simon Plouffe, Log(2*pi) to 2000 places

Cf. A002162, A053510, A094642, A131659.

RealDigits[N[Log[2*Pi], 100]][[1]] (* Arkadiusz Wesolowski, Aug 29 2011 *)

{ default(realprecision, 20080); x=log(2*Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b061444.txt", n, " ", d)) } \\ Harry J. Smith, Jul 22 2009

Equals A002162 + A053510 = A131659 - A094642. - R. J. Mathar, Aug 27 2011

Equals 1 + Sum_{k>=1} zeta(2*k)/(k*(2*k + 1)). - Amiram Eldar, Aug 20 2020

A152814 Decimal expansion of log_19 (2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

			.23540891336663823644696546003684185383540677662781604728813...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_19(m): this sequence, A153027 (m=3), A153117 (m=4), A153451 (m=5), A153609 (m=6), A153629 (m=7), A153871 (m=8), A154018 (m=9), A154169 (m=10), A154190 (m=11), A154211 (m=12), A154401 (m=13), A154491 (m=14), A154697 (m=15), A154837 (m=16), A154899 (m=17), A154975 (m=18), A155531 (m=20), A155686 (m=21), A155787 (m=22), A155906 (m=23), A156000 (m=24).

RealDigits[Log[19, 2], 10, 100][[1]] (* Vincenzo Librandi, Sep 05 2013 *)

log(2)/log(19) \\ Charles R Greathouse IV, Aug 06 2020

Equals A002162 / A016642.

A152821 Decimal expansion of log_20 (2).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 30 2009

			.23137821315975917426369770109764049589096550737356586183961...

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for transcendental numbers

Cf. decimal expansion of log_20(m): this sequence, A153035 (m=3), A153124 (m=4), A153454 (m=5), A153610 (m=6), A153630 (m=7), A153872 (m=8), A154019 (m=9), A154170 (m=10), A154191 (m=11), A154212 (m=12), A154433 (m=13), A154492 (m=14), A154705 (m=15), A154838 (m=16), A154900 (m=17), A154976 (m=18), A155115 (m=19), A155687 (m=21), A155789 (m=22), A155907 (m=23), A156015 (m=24).

RealDigits[Log[20, 2], 10, 100][[1]] (* Vincenzo Librandi, Sep 05 2013 *)

log(2)/log(20) \\ Charles R Greathouse IV, Aug 06 2020

Equals A002162 / A016643 = 1/A155172. - R. J. Mathar, Aug 13 2024

A038608 a(n) = n*(-1)^n.

Original entry on oeis.org

0, -1, 2, -3, 4, -5, 6, -7, 8, -9, 10, -11, 12, -13, 14, -15, 16, -17, 18, -19, 20, -21, 22, -23, 24, -25, 26, -27, 28, -29, 30, -31, 32, -33, 34, -35, 36, -37, 38, -39, 40, -41, 42, -43, 44, -45, 46, -47, 48, -49, 50, -51, 52, -53, 54, -55, 56, -57, 58, -59, 60, -61, 62, -63, 64, -65

Offset: 0

Vasiliy Danilov (danilovv(AT)usa.net), Jul 1998

a(n) is the determinant of the (n+1) X (n+1) matrix with 0's in the main diagonal and 1's elsewhere. - Franz Vrabec, Dec 01 2007
Sum_{n>0} 1/a(n) = -log(2). - Jaume Oliver Lafont, Feb 24 2009
Pisano period lengths:  1, 2, 6, 4, 10, 6, 14, 8, 18, 10, 22, 12, 26, 14, 30, 16, 34, 18, 38, 20, ... (is this A066043?). - R. J. Mathar, Aug 10 2012
a(n) is the determinant of the (n+1) X (n+1) matrix whose i-th row, j-th column entry is the value of the cubic residue symbol ((j-i)/p) where p is a prime of the form 3k+2 and n < p. - Ryan Wood, Nov 09 2017
a(n-1) is the difference in the number of even minus odd parity derangements (permutations with no fixed points) in symmetric group S_n. - Julian Hatfield Iacoponi, Aug 01 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Tanya Khovanova, Recursive Sequences
László Németh, The trinomial transform triangle, J. Int. Seqs., Vol. 21 (2018), Article 18.7.3. Also arXiv:1807.07109 [math.NT], 2018.
Index entries for linear recurrences with constant coefficients, signature (-2,-1).

Cf. A002162 (log(2)).
Cf. A001477.
Cf. A003221, A000387 (even, odd derangements).

a038608 n = n * (-1) ^ n
a038608_list = [0, -1] ++ map negate
   (zipWith (+) a038608_list (map (* 2) $ tail a038608_list))
-- Reinhard Zumkeller, Nov 24 2012

[n*(-1)^n: n in [0..80]]; // Vincenzo Librandi, Jun 08 2011

A038608 := n->n*(-1)^n; seq(A038608(n), n=0..100);

Array[# (-1)^# &, 66, 0] (* Michael De Vlieger, Nov 18 2017 *)
Table[If[EvenQ[n],n,-n],{n,0,70}] (* Harvey P. Dale, Jan 17 2022 *)

a(n)=n*(-1)^n \\ Charles R Greathouse IV, Dec 07 2011

def A038608(n): return -n if n&1 else n # Chai Wah Wu, Nov 14 2022

G.f.: -x/(1+x)^2.
E.g.f: -x*exp(-x).
a(n) = -2*a(n-1) - a(n-2) for n >= 2. - Jaume Oliver Lafont, Feb 24 2009
a(n) = A003221(n+1)-A000387(n+1). - Julian Hatfield Iacoponi, Aug 01 2024

Edited by Frank Ellermann, Jan 28 2002

A016655 Decimal expansion of log(32) = 5*log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

The function exp(x) has its maximum curvature where x = -(1/10)*5*log(2) = -log(2)/2 = 0.34657... - Dimitri Papadopoulos, Oct 27 2022

This maximum curvature occurs at the point with coordinates (x_M = -log(2)/2 = -(this constant)/10; y_M = sqrt(2)/2 = A010503) and is equal to 2*sqrt(3)/9 = A212886. - Bernard Schott, Dec 23 2022

			3.465735902799726547086160607290882840377500671801276270603400047466968...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Harry J. Smith, Table of n, a(n) for n = 1..20000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
O. Espinosa, V. H. Moll, On some integrals involving the Hurwitz Zeta function: Part 1, Raman. J. 6 (2002) 159-188, eq. (5.7)
Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2021.
R. S. Melham and A. G. Shannon, Inverse Trigonometric Hyperbolic Summation Formulas Involving Generalized Fibonacci Numbers, The Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 32-40.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), chap 7.1
H. M. Srivastava, M. L. Glasser, V. S. Adamchik, Some definite integrals associated with the Riemann Zeta Function, Z. Anal. Anw. 19 (3) (2000) 831-846
Index to sequences related to curves.
Index entries for transcendental numbers.

Cf. A000045, A000032, A060851, A195909, A195913, A195697, A016460 (continued fraction).

Cf. A010503, A212886.

[5*Log(2)]; // Vincenzo Librandi, Jan 02 2016

RealDigits[5 N [Log[2], 100]] [[1]] (* Vincenzo Librandi, Jan 02 2016 *)

log(32) \\ Charles R Greathouse IV, Jan 24 2012

log(2)/2 = (1 - 1/2 - 1/4) + (1/3 - 1/6 - 1/8) + (1/5 - 1/10 - 1/12) + ... [Jolley, Summation of Series, Dover (1961) eq (73)]
Equals 10*log(2)/2 = 5*log(2) = 5*A002162, so 10*(1/2 - 1/4 + 1/6 - 1/8 + 1/10 - ... + (-1)^(k+1)/(2*k) + ...) = log(32). - Eric Desbiaux, Nov 26 2008
-log(2)/2 = lim_{n->oo} ((Sum_{k=2..n} arctanh(1/k)) - log(n)). - Jean-François Alcover, Aug 07 2014, after Steven Finch
Equals log(sqrt(2)) with offset 0. - Michel Marcus, Feb 19 2017
Equals (5/4)*Sum_{k=1..4} (-1)^(k+1) gamma(0, k/4) where gamma(n,x) denotes the generalized Stieltjes constants. - Peter Luschny, May 16 2018
From Amiram Eldar, Jun 29 2020: (Start)
log(2)/2 = arctanh(1/3) = arcsinh(1/sqrt(8)).
log(2)/2 = Integral_{x=0..Pi/4} tan(x) dx.
log(2)/2 = Sum_{k>=0} (-1)^k/(2*k+2).
log(2)/2 = Sum_{k>=1} 1/A060851(k). (End)
log(2)/2 = Sum_{k>=1} (-1)^(k+1) * arctanh(Lucas(2*k+3)/Fibonacci(2*k+3)^2) (Melham and Shannon, 1995). - Amiram Eldar, Jan 15 2022
Equals 10 * Integral_{1..oo} dx/(x*(1+x^2)). [Nahin] - R. J. Mathar, May 22 2024
Equals -10*Integral_{q=0..1} q*log(sin(Pi*q))dq. [Espinosa] - R. J. Mathar, Aug 13 2024
log(2)/2 = Sum_{k>=2} (-1)^(k) * arccoth(k). - Antonio Graciá Llorente, Sep 16 2024
-0.34657359... = Sum_{k>=0} zeta(2k)/((2k+1)*2^(2k)), [Srivastava (2.20)] - R. J. Mathar, Feb 12 2020
Equals 10*Integral_{x=0..1} Ei((1 + sqrt(2))*log(x)) - li(x) dx, where Ei is the exponential integral and li is the logarithmic integral. - Kritsada Moomuang, Jun 06 2025

A020860 Decimal expansion of log(7)/log(2).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			2.807354922...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
L. Adleman, Molecular Computation of Solutions to Combinatorial Problems, Science 266 (1994): 1021-1024.
Aran Nayebi, Fast matrix multiplication techniques based on the Adleman-Lipton model, arXiv:0912.0750 [q-bio.QM], 2009-2011.
D. K. Nguyen, I. Lavallée, and M. Bui, A New Direction to Parallelize Winograd's Algorithm on Distributed Memory Computers, Modeling, Simulation and Optimization of Complex Processes Proceedings of the Third International Conference on High Performance Scientific Computing, March 6-10, 2006, Hanoi, Vietnam: 445-457.
V. Pan, How can we speed up matrix multiplication?, SIAM Review, 26 (1984): 393-416.
S. Robinson, Toward an Optimal Algorithm for Matrix Multiplication, SIAM News 38 (2005): 1-3.
V. Strassen, Gaussian elimination is not optimal, Numer. Math. 13 (1969): 354-356. MR 40:2223.
Yahoo Questions, Prove log(base 2)7 is rational?
Index entries for transcendental numbers

Cf. A002162, A016630.

RealDigits[Log[2, 7], 10, 100][[1]] (* Vincenzo Librandi, Aug 29 2013 *)

log(7)/log(2) \\ Charles R Greathouse IV, May 15 2019

A244009 Decimal expansion of 1 - log(2).

Original entry on oeis.org

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

Offset: 0

Franklin T. Adams-Watters, Jun 17 2014

Fraction of numbers which are sqrt-smooth, see A048098 and A063539. - Charles R Greathouse IV, Jul 14 2014

Asymptotic survival probability in the 100 prisoners problem. - Alois P. Heinz, Jul 08 2022

			0.30685281944005469058276787854...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, pp. 43-44.

Peter Bala, A continued fraction expansion for the constant 1 - log(2)
Donald E. Knuth and Luis Trabb Pardo, Analysis of a simple factorization algorithm, Theoretical Computer Science 3:3 (1976), pp. 321-348.
The Ramanujan Machine, Using algorithms to discover new mathematics.
Wikipedia, 100 prisoners problem
Index entries for transcendental numbers

Cf. A002162, A019739, A024168, A100381, A135002, A188859.

Essentially the same digits as A239354.

f:= sum(1/(2*k*(2*k+1)), k=1..infinity):
s:= convert(evalf(f, 140), string):
seq(parse(s[i+1]), i=1..106);  # Alois P. Heinz, Jun 17 2014

RealDigits[1-Log[2],10,120][[1]] (* Harvey P. Dale, Sep 23 2016 *)

1-log(2) \\ Charles R Greathouse IV, Jul 14 2014

Equals Sum_{k>=0} 1/(2*k*(2*k+1)) = A239354 + 1/4 = A188859/2.
From Amiram Eldar, Aug 07 2020: (Start)
Equals Sum_{k>=1} 1/(k*(k+1)*2^k) = Sum_{k>=2} 1/A100381(k).
Equals Sum_{k>=2} (-1)^k * zeta(k)/2^k.
Equals Integral_{x=1..oo} 1/(x^2 + x^3) dx. (End)
Equals log(e/2) = log(A019739) = -log(2/e) = -log(A135002). - Wolfdieter Lang, Mar 04 2022
Equals lim_{n->oo} A024168(n)/n!. - Alois P. Heinz, Jul 08 2022
Equals 1/(4 - 4/(7 - 12/(10 - ... - 2*n*(n-1)/((3*n+1) - ...)))) (an equivalent continued fraction for 1 - log(2) was conjectured by the Ramanujan machine). - Peter Bala, Mar 04 2024
Equals Sum_{k>=1} zeta(2*k)/((2*k + 1)*2^(2*k-1)) (see Finch). - Stefano Spezia, Nov 02 2024

A073000 Decimal expansion of arctangent of 1/2.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 03 2002

The angle at which you must shoot a cue ball on a standard pool table so that it will strike all four sides and return to its origin. [Barrow] - Robert G. Wilson v, Nov 29 2015

			Arctan(1/2)
=0.463647609000806116214256231461214402028537054286120263810933088720197864165... radians
=26°.56505117707798935157219372045329467120421429964522102798601631528806582148474...
=26°33'.9030706246793610943316232271976802722528579787132616791609789172839492890...
=26°33'54".184237480761665659897393631860816335171478722795700749658735037036957...
complement = 63°.43494882292201064842780627954670532879578570035477897201398368471...
supplement = 153°.4349488229220106484278062795467053287957857003547789720139836847...

John D. Barrow, One Hundred Essential Things You Didn't Know You Didn't Know, W. W. Norton & Co., NY & London, 2008.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 242.

G. C. Greubel, Table of n, a(n) for n = 0..5000
Peter Bala, New series for old functions.
R. J. Mathar, Hierarchical Subdivision of the Simple Cubic Lattice, arXiv preprint arXiv:1309.3705 [math.MG], 2013.
Simon Plouffe, arctan(1/2).
Index entries for transcendental numbers

Cf. A001622, A002162, A002391, A105531, A254619.

evalf(arctan(0.5)) ; # R. J. Mathar, Aug 22 2013

Mathematica
```
RealDigits[ ArcTan[1/2], 10, 110] [[1]]
```

default(realprecision,2000); atan(1/2) \\ Anders Hellström, Nov 30 2015

Equals Pi/2 - A105199 = A019669-A105199. - R. J. Mathar, Aug 21 2013
From Peter Bala, Feb 04 2015: (Start)
Arctan(1/2) = 1/2*Sum_{k >= 0} (-1)^k/((2*k + 1)*4^k).
Define a pair of integer sequences A(n) = 4^n*(2*n + 1)!/n! and B(n) = A(n)*Sum_{k = 0..n} (-1)^k/((2*k + 1)*4^k). Both sequences satisfy the same second order recurrence equation u(n) = (12*n + 10)*u(n-1) + 16*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 2*arctan(1/2) = 1 - 2/(24 + 16*3^2/(34 + 16*5^2/(46 + ... + 16*(2*n - 1)^2/((12*n + 10) + ...)))). See A002391, A105531 and A002162 for similar expansions.
Arctan(1/2) = 2/5 * Sum_{k >= 0} (4/5)^k/((2*k + 1)*binomial(2*k,k)).
Define a pair of integer sequences C(n) = 5^n*(2*n + 1)!/n! and D(n) = C(n)*Sum_{k = 0..n} (4/5)^k/((2*k + 1)*binomial(2*k,k)). Both sequences satisfy the same second order recurrence equation u(n) = (24*n + 10)*u(n-1) - 40*n*(2*n - 1)^2*u(n-2). From this observation we obtain the continued fraction expansion 5/2*arctan(1/2) = 1 + 4/(30 - 240/(58 - 600/(82 - ... - 40*n*(2*n - 1)/((24*n + 10) - ... )))).
Arctan(1/2) = 2/25 * Sum_{k >= 0} (24*k + 17)*(4/5)^(2*k)/( (4*k + 1)*(4*k + 3)*binomial(4*k,2*k) ).
Arctan(1/2) = 2/125 * Sum_{k >= 0} (1116*k^2 + 1446*k + 433)*(4/5)^(3*k)/( (6*k + 1)*(6*k + 3)*(6*k + 5)*binomial(6*k,3*k) ). (End)
Equals Integral_{x = 0..oo} exp(-2*x)*sin(x)/x dx. - Peter Bala, Nov 05 2019
Equals 2 * arccot(phi^3), where phi is the golden ratio (A001622). - Amiram Eldar, Jul 06 2023
Equals Sum_{n >= 1} i/(n*P(n, 2*i)*P(n-1, 2*i)) = (1/2)*Sum_{n >= 1} (-1)^(n+1)*4^n/(n*A098443(n)*A098443(n-1)), where i = sqrt(-1) and P(n, x) denotes the n-th Legendre polynomial. The n-th summand of the series is O( 1/(3 + 2*sqrt(2))^n ). - Peter Bala, Mar 16 2024

A228725 Decimal expansion of the generalized Euler constant gamma(1,2).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Aug 31 2013

The complement (A239097) is gamma(0,2) = lim_{x->oo} ((Sum_{n=1..x, n even} 1/n) - log(x)/2) = (A001620 - A002162)/2 = -0.05796575... - R. J. Mathar, Sep 06 2013

			0.63518142273073908501187210577028949955882973515008942646322...

G. C. Greubel, Table of n, a(n) for n = 0..10000
M. L. Glasser, A note on Beukers's and related integrals, Amer. Math. Monthly 126(4) (2019), 361-363.
J. C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013. See Section 3.8.
D. H. Lehmer, Euler constants for arithmetic progressions, Acta Arith. 27 (1975), 125-142.

Cf. A001620, A002162, A020759, A239097.

SetDefaultRealField(RealField(100)); R:= RealField();
(EulerGamma + Log(2))/2; // G. C. Greubel, Aug 27 2018

Maple
```
(gamma+log(2))/2 ; evalf(%) ;
```

RealDigits[(EulerGamma+Log[2])/2,10,120][[1]] (* Harvey P. Dale, Dec 26 2013 *)

(Euler+log(2))/2 \\ Charles R Greathouse IV, Jul 21 2015

Equals lim_{x->oo} ((Sum_{n=1..x, n odd} 1/n) - log(x)/2).
Equals (A001620 + A002162)/2.
From Amiram Eldar, Jun 30 2020: (Start)
Equals -Integral_{x=0..1} log(log(1/x))*x dx.
Equals -Integral_{x=0..oo} exp(-2*x)*log(x) dx. (End)
Equals Integral_{x=0..1, y=0..1} log(-log(x*y))*x*y/log(x*y) dx dy. (Apply Theorem 1 or Theorem 2 of Glasser (2019) to one of Amiram Eldar's integrals.) - Petros Hadjicostas, Jun 30 2020
Equals -(psi(1/2) + log(2))/2 = (A020759 - A002162)/2. - Amiram Eldar, Jan 07 2024

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