A006752 -id:A006752 - cp's OEIS frontend

A175571 Decimal expansion of the Dirichlet beta function of 5.

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

Offset: 0

R. J. Mathar, Jul 15 2010

The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547.

			0.99615782807708806400631936...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(5) ; x := evalf(x) ;

RealDigits[ DirichletBeta[5], 10, 105] // First (* Jean-François Alcover, Feb 20 2013, updated Mar 14 2018 *)

5*Pi^5/1536 \\ Charles R Greathouse IV, Jan 31 2018

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(5) \\ Charles R Greathouse IV, Jan 31 2018

Equals 5*Pi^5/1536 = Sum_{n>=1} A101455(n)/n^5, where Pi^5 = A092731. [corrected by R. J. Mathar, Feb 01 2018]
Equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^5)^(-1). - Amiram Eldar, Nov 06 2023

A175572 Decimal expansion of the Dirichlet beta function of 4.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 15 2010

This is the value of the Dirichlet L-series for A101455 at s=4, see arXiv:1008.2547, L(m=4,r=2,s=4).

			0.988944551741105336108422633...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (308).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Mark W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064, proposition 5.
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(4) ; x := evalf(x) ;

RealDigits[ DirichletBeta[4], 10, 105] // First (* Jean-François Alcover, Feb 11 2013, updated Mar 14 2018 *)

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(4) \\ Charles R Greathouse IV, Jan 31 2018

Equals Sum_{n>=1} A101455(n)/n^4. [corrected by R. J. Mathar, Feb 01 2018]
Equals (PolyGamma(3, 1/4) - PolyGamma(3, 3/4))/1536. - Jean-François Alcover, Jun 11 2015
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^4)^(-1). - Amiram Eldar, Nov 06 2023

A248938 Decimal expansion of beta = G^2(2/3)Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Oct 17 2014

			0.4612609086138613...

Steven R. Finch, Apollonian circles with integer curvatures, p. 6. [Cached copy, with permission of the author]
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 256.
Elena Fuchs and Katherine Sanden, Some experiments with integral Apollonian circle packings, arXiv:1001.1406 [math.NT] p. 7.
Peter Sarnak, Integral Apollonian Packings, Princeton MAA Lecture - January, 2009, p. 21.

Cf. A006752, A042944, A042946, A052483, A189226, A189227, A248930.

kmax = 25; Clear[P]; Do[P[k] = Product[p = Prime[n]; If[Mod[p, 4] == 3 , 1 - 2/(p*(p - 1)^2) // N[#, 40]&, 1], {n, 1, 2^k}]; Print["P(", k, ") = ", P[k]], {k, 10, kmax}]; beta = Catalan^2*(2/3)*P[kmax]; RealDigits[beta, 10, 16] // First
(* -------------------------------------------------------------------------- *)
$MaxExtraPrecision = 1000; digits = 121;
f[p_] := (1 - 2/(p*(p - 1)^2));
coefs = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, 1000}], x]];
S[m_, n_, s_] := (t = 1; sums = 0; difs = 1; While[Abs[difs] > 10^(-digits - 5) || difs == 0, difs = (MoebiusMu[t]/t) * Log[If[s*t == 1, DirichletL[m, n, s*t], Sum[Zeta[s*t, j/m]*DirichletCharacter[m, n, j]^t, {j, 1, m}]/m^(s*t)]]; sums = sums + difs; t++]; sums);
P[m_, n_, s_] := 1/EulerPhi[m] * Sum[Conjugate[DirichletCharacter[m, r, n]] * S[m, r, s], {r, 1, EulerPhi[m]}] + Sum[If[GCD[p, m] > 1 && Mod[p, m] == n, 1/p^s, 0], {p, 1, m}];
m = 2; sump = 0; difp = 1; While[Abs[difp] > 10^(-digits - 5) || difp == 0, difp = coefs[[m]]*P[4, 3, m]; sump = sump + difp; m++];
RealDigits[Chop[N[2*Catalan^2/3 * Exp[sump], digits]], 10, digits - 1][[1]] (* Vaclav Kotesovec, Jan 16 2021 *)

beta = (G^2/3)*A248930, where G is Catalan's constant A006752.

More digits from Vaclav Kotesovec, Jun 27 2020

A016826 a(n) = (4n + 2)^2.

Original entry on oeis.org

4, 36, 100, 196, 324, 484, 676, 900, 1156, 1444, 1764, 2116, 2500, 2916, 3364, 3844, 4356, 4900, 5476, 6084, 6724, 7396, 8100, 8836, 9604, 10404, 11236, 12100, 12996, 13924, 14884, 15876, 16900, 17956

Offset: 0

N. J. A. Sloane

A bisection of A016742. Sequence arises from reading the line from 4, in the direction 4, 36, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008

Harvey P. Dale, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Equals A001539 + 1.

Cf. A000290, A010503, A016742, A016754, A016802, A016814, A016838.

A016826:=n->(4*n + 2)^2; seq(A016826(n), n=0..40); # Wesley Ivan Hurt, Feb 24 2014

(4*Range[0,40]+2)^2 (* or *) LinearRecurrence[{3,-3,1},{4,36,100},40] (* Harvey P. Dale, Nov 24 2011 *)

a(n)=(4*n+2)^2 \\ Charles R Greathouse IV, Oct 07 2015

a(n) = a(n-1) + 32*n (with a(0)=4). - Vincenzo Librandi, Dec 15 2010
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3), with a(0)=4, a(1)=36, a(2)=100. - Harvey P. Dale, Nov 24 2011
G.f.: -((4*(x^2+6*x+1))/(x-1)^3). - Harvey P. Dale, Nov 24 2011
a(n) = A000290(A016825(n)). - Wesley Ivan Hurt, Feb 24 2014
From Amiram Eldar, Jun 28 2020: (Start)
Sum_{n>=0} 1/a(n) = Pi^2/32.
Sum_{n>=0} (-1)^n/a(n) = G/4, where G is the Catalan constant (A006752). (End)
From Amiram Eldar, Jan 29 2021: (Start)
Product_{n>=0} (1 + 1/a(n)) = cosh(Pi/4).
Product_{n>=0} (1 - 1/a(n)) = 1/sqrt(2) (A010503). (End)

A016838 a(n) = (4n + 3)^2.

Original entry on oeis.org

9, 49, 121, 225, 361, 529, 729, 961, 1225, 1521, 1849, 2209, 2601, 3025, 3481, 3969, 4489, 5041, 5625, 6241, 6889, 7569, 8281, 9025, 9801, 10609, 11449, 12321, 13225, 14161, 15129, 16129, 17161, 18225

Offset: 0

N. J. A. Sloane

If Y is a fixed 2-subset of a (4n+1)-set X then a(n-1) is the number of 3-subsets of X intersecting Y. - Milan Janjic, Oct 21 2007
A bisection of A016754. Sequence arises from reading the line from 9, in the direction 9, 49, ... in the square spiral whose vertices are the squares A000290. - Omar E. Pol, May 24 2008
Using (n,n+1) to generate a Pythagorean triangle with sides of lengths xJ. M. Bergot, Jul 17 2013

Vincenzo Librandi, Table of n, a(n) for n = 0..860
Milan Janjic, Two Enumerative Functions
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000290, A001539, A016742, A016754, A016802, A016814, A016826, A068465, A087197.

[(4*n+3)^2: n in [0..50]]; // Vincenzo Librandi, Apr 26 2011

A016838:=n->(4*n + 3)^2; seq(A016838(n), n=0..50); # Wesley Ivan Hurt, Feb 24 2014

Table[(4*n + 3)^2, {n, 0, 40}] (* Vaclav Kotesovec, Nov 14 2017 *)

a(n)=(4*n+3)^2 \\ Charles R Greathouse IV, Jun 17 2017

Denominators of first differences Zeta[2,(4n-1)/4]-Zeta[2,(4(n+1)-1)/4]. - Artur Jasinski, Mar 03 2010
From George F. Johnson, Oct 03 2012: (Start)
G.f.: (9+22*x+x^2)/(1-x)^3.
a(n+1) = a(n) + 16 + 8*sqrt(a(n)).
a(n+1) = 2*a(n) - a(n-1) + 32 = 3*a(n) - 3*a(n-1) + a(n-2).
a(n-1)*a(n+1) = (a(n)-16)^2; a(n+1) - a(n-1) = 16*sqrt(a(n)).
a(n) = A016754(2*n+1) = (A004767(n))^2.
(End)
Sum_{n>=0} 1/a(n) = Pi^2/16 - G/2, where G is the Catalan constant (A006752). - Amiram Eldar, Jun 28 2020
Product_{n>=0} (1 - 1/a(n)) = Gamma(3/4)^2/sqrt(Pi) = A068465^2 * A087197. - Amiram Eldar, Feb 01 2021

A243381 Decimal expansion of Pi^2/(16K^2G) = Product_{p prime congruent to 3 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.

Original entry on oeis.org

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

Offset: 1

Jean-François Alcover, Jun 04 2014

			1.1530805615854478703652580685617633651...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constant, p. 101.

G. C. Greubel, Table of n, a(n) for n = 1..2500
Eric Weisstein's MathWorld, Ramanujan constant.
Eric Weisstein's MathWorld, Catalan's Constant.

Cf. A002145, A064533, A243379.

digits = 101; LandauRamanujanK = 1/Sqrt[2]*NProduct[((1 - 2^(-2^n))*Zeta[2^n]/DirichletBeta[2^n])^(1/2^(n + 1)), {n, 1, 24}, WorkingPrecision -> digits + 5]; Pi^2/(16*LandauRamanujanK^2*Catalan) // RealDigits[#, 10, digits] & // First (* updated Mar 14 2018 *)

Equals Pi^2/(16*K^2*G), where K is the Landau-Ramanujan constant (A064533) and G Catalan's constant (A006752).

A243380 * A243381 = 12/Pi^2. - Vaclav Kotesovec, Apr 30 2020

A245058 Decimal expansion of the real part of Li_2(I), negated.

Original entry on oeis.org

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

Offset: 0

Robert G. Wilson v, Aug 21 2014

This is the decimal expansion of the real part of the dilogarithm of the square root of -1. The imaginary part is Catalan's number (A006752).

5*Pi^2/24 = 10 * (this constant) equals the asymptotic mean of the abundancy index of the even numbers. - Amiram Eldar, May 12 2023

			0.2056167583560283045590518958307531486523687376508498047169447786712509338004...

G. C. Greubel, Table of n, a(n) for n = 0..10000 (terms 0..104 from Robert G. Wilson v)
R. J. Mathar, Some definite integrals over a power multiplied by four modified Bessel functions vixra:1606.0141 (2016) eq. (39)
Paul J. Nahin, Inside interesting integrals, Undergrad. Lecture Notes in Physics, Springer (2020), (6.3.13)

Cf. A006752, A007949, A072691, A222171, A242301.

SetDefaultRealField(RealField(100)); R:=RealField(); Pi(R)^2/48; // G. C. Greubel, Aug 25 2018

RealDigits[ Re[ PolyLog[2, I]], 10, 111][[1]] (* or *) RealDigits[ Zeta[2]/8, 10, 111][[1]] (* or *) RealDigits[ Pi^2/48, 10, 111][[1]]

zeta(2)/8 \\ Charles R Greathouse IV, Aug 27 2014

(pi**2/48).n(200) # F. Chapoton, Mar 16 2020

Also equals -zeta(2)/8 = -Pi^2/48.
Also equals the Bessel moment Integral_{0..inf} x I_1(x) K_0(x)^2 K_1(x) dx. - Jean-François Alcover, Jun 05 2016
From Terry D. Grant, Sep 11 2016: (Start)
Equals Sum_{n>=0} (-1)^n/(2n+2)^2.
Equals (Sum_{n>=1} 1/(2n)^2)/2 = A222171/2. (End)
Equals Sum_{k>=1} A007949(k)/k^2. - Amiram Eldar, Jul 13 2020
Equals a tenth of integral_0^{pi/2} arccos[cos x/(1+2 cos x)]dx [Nahin]. - R. J. Mathar, May 22 2024
Equals Integral_{x>=0} x/(exp(2*x) + 1) dx. - Kritsada Moomuang, May 29 2025

A087784 Number of solutions to x^2 + y^2 + z^2 = 1 mod n.

Original entry on oeis.org

1, 4, 6, 24, 30, 24, 42, 96, 54, 120, 110, 144, 182, 168, 180, 384, 306, 216, 342, 720, 252, 440, 506, 576, 750, 728, 486, 1008, 870, 720, 930, 1536, 660, 1224, 1260, 1296, 1406, 1368, 1092, 2880, 1722, 1008, 1806, 2640, 1620, 2024, 2162, 2304, 2058, 3000

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Michael De Vlieger, Table of n, a(n) for n = 1..10000
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, Journal of Integer Sequences, 17 (2014), Article 14.11.6.

Cf. A006752, A060968, A087687, A208895, A264083.

Table[With[{f = FactorInteger[n][[All, 1]]}, Apply[Times, Map[1 + 1/# &, Select[f, Mod[#, 4] == 1 &]]] Apply[Times, Map[1 - 1/# &, Select[f, Mod[#, 4] == 3 &]]] (1 + Boole[Divisible[n, 4]]/2) n^2] - Boole[n == 1], {n, 50}] (* Michael De Vlieger, Feb 15 2018 *)

a(n) = {my(f=factor(n)); if ((n % 4), 1, 3/2)*n^2*prod(k=1, #f~, p = f[k,1]; m = p % 4; if (m==1, 1+1/p, if (m==3, 1-1/p, 1)));} \\ Michel Marcus, Feb 14 2018

a(n) = n^2 * (3/2 if 4|n) * Product_{primes p == 1 mod 4 dividing n} (1+1/p) * Product_{primes p == 3 mod 4 dividing n} (1-1/p). - Bjorn Poonen, Dec 09 2003

Sum_{k=1..n} a(k) ~ c * n^3 + O(n^2 * log(n)), where c = 36*G/Pi^4 = 0.338518..., where G is Catalan's constant (A006752) (Tóth, 2014). - Amiram Eldar, Oct 18 2022

More terms from David Wasserman, Jun 17 2005

A175570 Decimal expansion of the Dirichlet beta function of 6.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 15 2010

			0.998685222218438135441600...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(6) ; x := evalf(x) ;

RealDigits[ DirichletBeta[6], 10, 105] // First (* Jean-François Alcover, Feb 11 2013, updated Mar 14 2018 *)

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(6) \\ Charles R Greathouse IV, Jan 31 2018

sumpos(n=1,(12288*n^5 - 30720*n^4 + 33280*n^3 - 19200*n^2 + 5808*n - 728)/(16777216*n^12 - 100663296*n^11 + 270532608*n^10 - 429916160*n^9 + 449249280*n^8 - 324796416*n^7 + 166445056*n^6 - 60899328*n^5 + 15793920*n^4 - 2833920*n^3 + 334368*n^2 - 23328*n + 729),1) \\ Charles R Greathouse IV, Feb 01 2018

Equals Sum_{n>=1} A101455(n)/n^6. [see arxiv:1008.2547, L(m=4,r=2,s=6)] [corrected by R. J. Mathar, Feb 01 2018]
Equals (PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/491520. - Jean-François Alcover, Jun 11 2015
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^6)^(-1). - Amiram Eldar, Nov 06 2023

A014538 Continued fraction for Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...

Original entry on oeis.org

0, 1, 10, 1, 8, 1, 88, 4, 1, 1, 7, 22, 1, 2, 3, 26, 1, 11, 1, 10, 1, 9, 3, 1, 1, 1, 1, 1, 1, 2, 2, 1, 11, 1, 1, 1, 6, 1, 12, 1, 4, 7, 1, 1, 2, 5, 1, 5, 9, 1, 1, 1, 1, 33, 4, 1, 1, 3, 5, 3, 2, 1, 2, 1, 2, 1, 7, 6, 3, 1, 3, 3, 1, 1, 2, 1, 14, 1, 4, 4, 1, 2, 4, 1, 17, 4, 1, 14, 1, 1, 1, 12, 1

Offset: 0

Eric W. Weisstein

First 4,851,389,025 terms computed by Eric W. Weisstein, Aug 07 2013.

			C = 0.91596559417721901505... = 0 + 1/(1 + 1/(10 + 1/(1 + 1/(8 + ...))))

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. J. Fee, Computation of Catalan's constant using Ramanujan's formula, in Proc. Internat. Symposium on Symbolic and Algebraic Computation (ISSAC '90). 1990, pp. 157-160.
Eric Weisstein's World of Mathematics, Catalan's Constant Continued Fraction
G. Xiao, Contfrac
Index entries for continued fractions for constants

Cf. A006752 (decimal expansion of Catalan's constant).
Cf. A099789 (high water marks), A099790 (positions of high water marks).
Cf. A006752, A104338, A153069, A153070, A054543, A118323. - Stuart Clary, Dec 17 2008

R:= RealField(100); ContinuedFraction(Catalan(R)); // G. C. Greubel, Aug 23 2018

ContinuedFraction[Catalan, 100] (* G. C. Greubel, Aug 23 2018 *)

default(realprecision, 100); contfrac(Catalan) \\ G. C. Greubel, Aug 23 2018

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A175571 Decimal expansion of the Dirichlet beta function of 5.

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A175572 Decimal expansion of the Dirichlet beta function of 4.

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A248938 Decimal expansion of beta = G^2*(2/3)*Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.

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A016826 a(n) = (4n + 2)^2.

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A016838 a(n) = (4n + 3)^2.

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A243381 Decimal expansion of Pi^2/(16*K^2*G) = Product_{p prime congruent to 3 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.

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A245058 Decimal expansion of the real part of Li_2(I), negated.

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A248938 Decimal expansion of beta = G^2(2/3)Product_{prime p == 3 (mod 4)} (1 - 2/(p*(p-1)^2)) (where G is Catalan's constant), a constant related to the problem of integral Apollonian circle packings.

A243381 Decimal expansion of Pi^2/(16K^2G) = Product_{p prime congruent to 3 modulo 4} (1 + 1/p^2), where K is the Landau-Ramanujan constant and G Catalan's constant.