A068466 -id:A068466 - cp's OEIS frontend

A002161 Decimal expansion of square root of Pi.

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

Offset: 1

N. J. A. Sloane

Also Gamma(1/2). - Franklin T. Adams-Watters, Apr 07 2006
The integral of the Gaussian function exp(-x^2) over the real line. - Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008
Also equals the average distance between two points in two dimensions where coordinates are independent normally distributed random variables with mean 0 and variance 1. - Jean-François Alcover, Oct 31 2014, after Steven Finch
Also diameter of a sphere whose surface area equals Pi^2. More generally, the square root of x is also the diameter of a sphere whose surface area equals x*Pi. - Omar E. Pol, Nov 11 2018
Convergents of continued fractions: 7/4, 16/9, 23/13, 39/22, 257/145, 296/167, 8545/4821, ... - R. J. Mathar, Jan 29 2025

			1.7724538509055160272981674833411451827975494561223871282138...

George Boros and Victor H. Moll, Irresistible integrals, Cambridge University Press (2006), p. 190.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.5.4, p. 33.
W. E. Mansell, Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8, Cambridge Univ. Press, 1964, p. XVIII.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, page 413.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 40.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Donald Knuth, Why pi?, Christmas Tree lecture, Dec 06 2010 (video).
Index entries for sequences related to the number Pi.
Index entries for transcendental numbers.

Cf. A000796, A002388, A073006, A195907, A203145.

Cf. decimal expansions of Gamma(1/k): A073005 (k=3), A068466 (k=4), A175380 (k=5), A175379 (k=6), A220086 (k=7), A203142 (k=8).

R:= RealField(100); Sqrt(Pi(R));  // G. C. Greubel, Mar 10 2018

evalf(sqrt(Pi),120); # Muniru A Asiru, Nov 11 2018

RealDigits[N[Sqrt[Pi], 120]][[1]] (* Richard Chapling (r.chappers(AT)gmail.com), Jun 05 2008 *)

default(realprecision, 20080); x=sqrt(Pi); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b002161.txt", n, " ", d)); \\ Harry J. Smith, May 01 2009

Equals (1/2) * Sum_{n>=0} ((-1)^n * (4*n+1) * (1/8)^(n+1) * (2^(n+1))^3 * Gamma(n+1/2)^3 / Gamma(n+1)^3). - Alexander R. Povolotsky, Mar 25 2013
Equals Integral_{x=0..1} 1/sqrt(-log(x)) dx. - Jean-François Alcover, Apr 29 2013
Equals Sum_{k>=0} (k+1/2)!/(k+2)!. - Amiram Eldar, Jun 19 2023
Equals Integral_{x=0..oo} exp(-x)/sqrt(x) dx. - Michal Paulovic, Sep 24 2023
Equals Integral_{x=0..oo} 4/(exp(x^2)*(2*x^2 + 1)^2) dx. - Kritsada Moomuang, Jun 05 2025

More terms from Franklin T. Adams-Watters, Apr 07 2006

A068465 Decimal expansion of Gamma(3/4).

Original entry on oeis.org

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

Offset: 1

Benoit Cloitre, Mar 10 2002

			Gamma(3/4) = 1.225416702465177645129098303362890526851239248108070611...

Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 43, equation 43:4:14 at page 414.

G. C. Greubel, Table of n, a(n) for n = 1..20000
Russell J. Matheson, GAMMA(3/4) computed to 14550 digits.
Simon Plouffe, GAMMA(3/4) to 256 places, see p. 65.
Index to sequences related to the Gamma function

Cf. A000796, A002193, A053004, A063448, A068466, A069998.

SetDefaultRealField(RealField(105)); Gamma(3/4); // G. C. Greubel, Mar 11 2018

evalf(GAMMA(3/4)) ; # R. J. Mathar, Jan 10 2013

RealDigits[Gamma[3/4], 10, 100][[1]] (* G. C. Greubel, Mar 11 2018 *)

default(realprecision, 100); gamma(3/4) \\ G. C. Greubel, Mar 11 2018

Gamma(3/4) * A068466 = sqrt(2)*Pi = A063448. - R. J. Mathar, Jun 18 2006
Equals Integral_{x>=0} x^(-1/4)*exp(-x) dx. - Vaclav Kotesovec, Nov 12 2020
Equals (Pi/2)^(1/4) * sqrt(AGM(1,sqrt(2))) = sqrt(A069998 * A053004). - Amiram Eldar, Jun 12 2021

A084943 Decagorials: n-th polygorial for k=10.

Original entry on oeis.org

1, 1, 10, 270, 14040, 1193400, 150368400, 26314470000, 6104957040000, 1813172240880000, 670873729125600000, 302564051835645600000, 163384587991248624000000, 104075982550425373488000000, 77224379052415627128096000000, 66026844089815361194522080000000, 64442199831659792525853550080000000

Offset: 0

Daniel Dockery (peritus(AT)gmail.com), Jun 13 2003

Daniel Dockery, Polygorials, Special "Factorials" of Polygonal Numbers, preprint, 2003.

Cf. A006472, A001044, A000680, A068466, A084939, A084940, A084941, A084942, A084944, A085356.

a := n->n!/2^n*product(8*i+2,i=0..n-1); [seq(a(j),j=0..30)];

polygorial[k_, n_] := FullSimplify[ n!/2^n (k -2)^n*Pochhammer[2/(k -2), n]]; Array[polygorial[10, #] &, 14, 0] (* Robert G. Wilson v, Dec 26 2016 *)

a(n)=n!/2^n*prod(i=1,n,8*i-6) \\ Charles R Greathouse IV, Dec 13 2016

a(n) = polygorial(n, 10) = (A000142(n)/A000079(n))*A084948(n) = (n!/2^n)*Product_{i=0..n-1} (8*i+2) = (n!/2^n)*8^n*Pochhammer(1/4, n) = (n!/2)*4^n*Gamma(n+1/4)*sqrt(2)*Gamma(3/4)/Pi.
a(n) = Product_{k=1..n} k*(4k-3). - Daniel Suteu, Nov 01 2017
D-finite with recurrence a(n) -n*(4*n-3)*a(n-1)=0. - R. J. Mathar, May 02 2022
a(n) ~ 2^(2*n+1) * n^(2*n + 1/4) * Pi /(Gamma(1/4) * exp(2*n)). - Amiram Eldar, Aug 28 2025

A068467 Decimal expansion of (1/4)! = Gamma(5/4).

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Mar 10 2002

			0.906402477055477077982671288966918000748791920720...

G. C. Greubel, Table of n, a(n) for n = 0..10000
J. M. Borwein and I. J. Zucker, Fast evaluation of the gamma function for small rational fractions using complete elliptic integrals of the first kind, IMA Journal of Numerical Analysis, vol. 12, no. 4, pp. 519-526, 1992.
Greg Martin, A product of Gamma function values at fractions with the same denominator, arXiv:0907.4384v1 [math.CA], 24-July-2009
Albert Nijenhuis, Small Gamma Products with Simple Values, arXiv:0907.1689v1 [math.CA], 9-July-2009.
Raimundas Vidunas, Expressions for values of the gamma function, arXiv:math/0403510 [math.CA], 30-March-2004.
Wikipedia, Particular values of the Gamma function
Index to sequences related to the Gamma function

Cf. A202623.

SetDefaultRealField(RealField(100)); Gamma(5/4); // G. C. Greubel, Mar 11 2018

evalf(GAMMA(5/4)) ; # R. J. Mathar, Jan 10 2013

RealDigits[Gamma[5/4],10,120][[1]] (* Harvey P. Dale, Aug 23 2013 *)

PARI
```
gamma(5/4) \\ Altug Alkan, Sep 18 2016
```

2^(3/4)*(2/e^(16*Pi) + 1)* Pi^(3/4)/(2^(13/16)/(sqrt(2) - 1)^(1/4) + 2^(1/4) + 1) is a very good approximation (~88 digits) which becomes exact if you replace (2/e^(16*Pi) + 1) by EllipticTheta[3,0,exp(-(16*Pi))]. [R. W. Gosper, Posting to Math Fun Mailing List, Dec 27 2011.]
Equals A068466 /4 . - R. J. Mathar, Jan 10 2013
Also equals integral_{0..oo} exp(-x^4) dx. - Jean-François Alcover, Mar 29 2013
Equals 2^(-5/4)*Pi^(3/4)*Product_{k>=1} tanh(Pi*k/2). - Keshav Raghavan, Aug 25 2016

Removed leading zero and adjusted offset, R. J. Mathar, Feb 06 2009
Additional reference from Joerg Arndt, Dec 28 2011
Edited by N. J. A. Sloane, Dec 28 2011

A256166 Decimal expansion of log(Gamma(1/4)).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, Mar 17 2015

			1.288022524698077457370610440219717295925377565112860...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A068466 (Gamma(1/4)), A115252 (first Malmsten's integral).

Cf. decimal expansions of log(Gamma(1/k)): A155968 (k=2), A256165 (k=3), A256167 (k=5), A255888 (k=6), A256609 (k=7), A255306 (k=8), A256610 (k=9), A256612 (k=10), A256611 (k=11), A256066 (k=12), A256614 (k=16), A256615 (k=24), A256616 (k=48).

evalf(log(GAMMA(1/4)),120); # Vaclav Kotesovec, Mar 17 2015

RealDigits[Log[Gamma[1/4]],10,105][[1]] (* Vaclav Kotesovec, Mar 17 2015 *)

log(gamma(1/4)) \\ Michel Marcus, Mar 17 2015

A203142 Decimal expansion of Gamma(1/8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Dec 29 2011

			7.5339415987976119046992298412151336246104195881490759409831...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Raimundas Vidunas, Expressions for values of the Gamma function, arXiv:math/0403510 [math.CA], 2004.
Wikipedia, Particular values of the Gamma function: General rational arguments
Index to sequences related to the Gamma function

SetDefaultRealField(RealField(100)); Gamma(1/8); // G. C. Greubel, Mar 10 2018

RealDigits[Gamma[1/8], 10, 100][[1]] (* Bruno Berselli, Dec 13 2012 *)
RealDigits[Pi^(1/8) * 2^(17/8) * EllipticK[1/2]^(1/4) * EllipticK[3 - 2*Sqrt[2]]^(1/2), 10, 100][[1]] (* Vaclav Kotesovec, Apr 15 2024 *)

default(realprecision, 100); gamma(1/8) \\ G. C. Greubel, Jan 15 2017

this * A203144 * A231863 /2^(1/4) = A068466. - R. J. Mathar, Jan 15 2021

A257955 Decimal expansion of Gamma(1/Pi).

Original entry on oeis.org

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

Offset: 1

Iaroslav V. Blagouchine, May 14 2015

The reference gives an interesting product representation in terms of rational multiple of 1/Pi for Gamma(1/Pi).

			2.8112975146708616421227908037104816935281655223291765...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/Pi, Mathematics of Computation (AMS), 2015.

Cf. A257957, A257958, A257959, A002161, A073005, A068466, A175380, A175379, A220086, A203142, A256190, A256191, A256192, A203140, A203139, A203138, A203137.

Maple
```
evalf(GAMMA(1/Pi), 117);
```
Mathematica
```
RealDigits[Gamma[1/Pi], 10, 117][[1]]
```

default(realprecision, 117); gamma(1/Pi)

A091670 Decimal expansion of Gamma(1/4)^4/(4*Pi^3).

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Jan 27 2004

Watson's first triple integral.
This is also the value of F. Morley's series from 1902 Sum_{k=0..n} (risefac(k,1/2)/k!)^3 = hypergeometric([1/2,1/2,1/2],[1,1],1) with the rising factorial risefac(n,x). See A277232, also for the Hardy reference and a MathWorld link. - Wolfdieter Lang, Nov 11 2016
This constant is transcendental due to a result of Nesterenko, who proves that Gamma(1/4) is algebraically independent of Pi. - Charles R Greathouse IV, Aug 19 2025

			1.39320392968567685918424626032536824265748121751561787897...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.9, p. 324.

G. C. Greubel, Table of n, a(n) for n = 1..10000
A. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 256, 6.1.17 , p. 557, 15.1.26.
M. L. Glasser, I. J. Zucker, Extended Watson integrals for the cubic lattices, Proc. Nat. Acad. Sci., Vol. 74, No. 5 (1977), p. 1800-1801.
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.5.1)
Yu. V. Nesterenko, Modular functions and transcendence questions, Sbornik: Mathematics, Vol. 187, No. 9 (1996), pp. 1319-1348. (English translation)
Tito Piezas III, Watson's triple integrals.
Eric Weisstein's World of Mathematics, Watson's Triple Integrals.
I. J. Zucker, 70+years of the Watson integrals, J. Stat. Phys., Vol. 145, No. 3 (2011), pp. 591-612.
Index entries for transcendental numbers.

Cf. A091671, A091672, A277232, A293238 (inverse), A068466.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^4/(4*Pi(R)^3); // G. C. Greubel, Oct 26 2018

Pi/GAMMA(3/4)^4 ; evalf(%) ; # R. J. Mathar, Jun 17 2016

RealDigits[ N[ Gamma[1/4]^4/(4*Pi^3), 102]][[1]] (* Jean-François Alcover, Nov 12 2012, after Eric W. Weisstein *)

1/agm(sqrt(1/2),1)^2 \\ Charles R Greathouse IV, Mar 03 2016

From Joerg Arndt, Nov 27 2010: (Start)
Equals 1/agm(1,sqrt(1/2))^2.
Equals Gamma(1/4)^4 / (4*Pi^3) = Pi / (Gamma(3/4))^4 = hypergeom([1/2,1/2],[1],1/2)^2, see the two Abramowitz - Stegun references. (End)
Equals the square of A175574. Equals A000796/A068465^4. - R. J. Mathar, Jun 17 2016
Equals hypergeom([1/2,1/2,1/2],[1,1],1) - Wolfdieter Lang, Nov 12 2016
Equals Sum_{k>=0} binomial(2*k,k)^3/2^(6*k). - Amiram Eldar, Aug 26 2020

A064853 Decimal expansion of the Lemniscate constant.

Original entry on oeis.org

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

Offset: 1

Eric W. Weisstein, Sep 22 2001

			5.244115108584239620929679...

Harry J. Smith, Table of n, a(n) for n = 1..5000
Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 15.
Lorenz Milla, World record computation of Lemniscate Constant (2,000,000,000,000 digits)
Eric Weisstein's World of Mathematics, Lemniscate Constant.
Eric Weisstein's World of Mathematics, Lemniscate.
Index entries for transcendental numbers.

Cf. A002193, A019727, A062539, A068466, A085565, A093341.

SetDefaultRealField(RealField(100)); R:= RealField(); Gamma(1/4)^2/Sqrt(2*Pi(R)); // G. C. Greubel, Oct 07 2018

First@RealDigits[ N[ Gamma[ 1/4 ]^2/Sqrt[ 2 Pi ], 102 ] ]

{ allocatemem(932245000); default(realprecision, 5080); x=gamma(1/4)^2/sqrt(2*Pi); for (n=1, 5000, d=floor(x); x=(x-d)*10; write("b064853.txt", n, " ", d)); } \\ Harry J. Smith, Jun 20 2009

gamma(1/2)*gamma(1/4)/gamma(3/4) \\ Charles R Greathouse IV, Oct 29 2021

Equals Gamma(1/4)^2/sqrt(2*Pi). - G. C. Greubel, Oct 07 2018
Equals 2*A062539 = 4*A085565. - Amiram Eldar, May 04 2022
From Stefano Spezia, Sep 23 2022: (Start)
Equals 4*Integral_{x=0..Pi/2} 1/sqrt(2*(1 - (1/2)*sin(x)^2)) dx [Gauss, 1799] (see Faulhuber et al.).
Equals 2*sqrt(2)*A093341. (End)

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jan 17 2006

This sequence (its negated version) is also the decimal expansion of the first Malmsten integral int_{x=1..infinity} log(log(x))/(1 + x^2) dx = int_{x=0..1} log(log(1/x))/(1 + x^2) dx = int_{x=0..infinity} 0.5*log(x)/cosh(x) dx = int_{x=Pi/4..Pi/2} log(log(tan(x))) dx = (1/2)*Pi*log(2) + (3/4)*Pi*log(Pi) - Pi*log(Gamma(1/4)). - Iaroslav V. Blagouchine, Mar 29 2015

			0.26044280630098844554009386878972721953181917772314...

G. C. Greubel, Table of n, a(n) for n = 0..5000
I. V. Blagouchine, Rediscovery of Malmsten's integrals, their evaluation by contour integration methods and some related results, The Ramanujan Journal, Volume 35, Issue 1, pp. 21-110, 2014, DOI: 10.1007/s11139-013-9528-5. PDF file
Eric Weisstein's World of Mathematics, Vardi's Integral

Cf. A256127 (second Malmsten integral), A256128 (third Malmsten integral), A256129 (fourth Malmsten integral), A068466 (Gamma(1/4)), A256166 (log(Gamma(1/4))), A002162 (log 2), A053510 (log Pi).

RealDigits[-Pi/2*Log[Sqrt[2 Pi] Gamma[3/4]/Gamma[1/4]], 10, 111][[1]] (* Robert G. Wilson v, Dec 06 2014 *)

(-Pi*log((sqrt(2*Pi)*gamma(3/4))/gamma(1/4)))/2 \\ Michel Marcus, Dec 06 2014

Equals integral_[0..1] log(1/log(1/x))/(1+x^2) dx. - Jean-François Alcover, Jan 28 2015

cp's OEIS Frontend

A002161 Decimal expansion of square root of Pi.

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A068465 Decimal expansion of Gamma(3/4).

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A084943 Decagorials: n-th polygorial for k=10.

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A068467 Decimal expansion of (1/4)! = Gamma(5/4).

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A256166 Decimal expansion of log(Gamma(1/4)).

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A203142 Decimal expansion of Gamma(1/8).

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Formula

A257955 Decimal expansion of Gamma(1/Pi).

Views

A115252 Decimal expansion of -(Pilog((sqrt(2Pi)*Gamma(3/4))/Gamma(1/4)))/2.