cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A006752 Decimal expansion of Catalan's constant 1 - 1/9 + 1/25 - 1/49 + 1/81 - ...

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

Comments

Usually denoted by G.
With the k-th appended term being 2*3*...*(2+k-2)*2^k*(2^k-1)*Bern(k) / (2*k!*(J^(k+2-1))). Bern(k) is a Bernoulli number and J is a large number of the form 4n + 1. See equation 3:3:7 in Spanier and Oldham. - Harry J. Smith, May 07 2009

Examples

			0.91596559417721901505460351493238411077414937428167213426649811962176301977...

References

  • Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, pages 57, 554.
  • Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 53-59.
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
  • Jerome Spanier and Keith B. Oldham, An Atlas of Functions, 1987, equations 1:7:3, 3:3:7.

Links

Crossrefs

Cf. A014538, A104338, A153069, A153070, A054543, A118323, A294970/A294971.

Programs

  • Magma
    R:= RealField(100); Catalan(R); // G. C. Greubel, Aug 21 2018
  • Maple
    evalf(Catalan) ; # R. J. Mathar, Apr 09 2013
  • Mathematica
    nmax = 1000; First[RealDigits[Catalan, 10, nmax]] (* Stuart Clary, Dec 17 2008 *)
Integrate[ArcTan[x]/x, {x, 0, 1}] (* N. J. A. Sloane, May 03 2013 *)
N[Im[PolyLog[2, I]], 100] (* Peter Luschny, Oct 04 2019 *)
  • PARI
    { mydigits=20000; default(realprecision, mydigits+80); s=1.0; n=5*mydigits; j=4*n+1; si=-1.0; for (i=3, j-2, s+=si/i^2; si=-si; i++; ); s+=0.5/j^2; ttk=4.0; d=4.0*j^3; xk=2.0; xkp=xk; for (k=2, 100000000, term=(ttk-1)*ttk*xkp; xk++; xkp*=xk; if (k>2, term*=xk; xk++; xkp*=xk; ); term*=bernreal(k)/d; sn=s+term; if (sn==s, break); s=sn; ttk*=4.0; d*=(k+1)*(k+2)*j^2; k++; ); x=10*s; for (n=0, mydigits, d=floor(x); x=(x-d)*10; write("b006752.txt", n, " ", d)); } /* Beta(2) = 1 - 1/3^2 + 1/5^2 - ... - 1/(J-2)^2 + 1/(2*J^2) + 2*Bern(0)/(2*J^3) - 2*3*4*Bern(2)/J^5 + ...  */
  • PARI
    default(realprecision,1000+2); /* 1000 terms */
s=sumalt(n=0,(-1)^n/(2*n+1)^2);
v=Vec(Str(s)); /* == ["0", ".", "9", "1", "5", "9", "6", ...*/
vector(#v-2,n,eval(v[n+2]))
/* Joerg Arndt, Aug 25 2011 */
  • PARI
    Catalan \\ Charles R Greathouse IV, Nov 20 2011
  • PARI
    (zetahurwitz(2,1/4)-Pi^2)/8 \\ Charles R Greathouse IV, Jan 30 2018
  • PARI
    lerchphi(-1, 2, 1/2)/4 \\ Charles R Greathouse IV, Jan 30 2025

Formula

G = Integral_{x=0..1} arctan(x)/x dx.
G = Integral_{x=0..1} 3*arctan(x*(1-x)/(2-x))/x dx. - Posting to Number Theory List by James Mc Laughlin, Sep 27 2007
G = (zeta(2,1/4)- zeta(2,3/4))/16. - Gerry Martens, May 27 2011 [With the Hurwitz zeta function zeta.]
G = (1/2)*Sum_{n>=0} (-1)^n * ((3*n+2)*8^n) / ((2*n+1)^3*C(2*n,n)^3) (from the Lima 2012 reference).
G = (-1/64)*Sum_{n>=1} (-1)^n * (2^(8*n) * (40*n^2-24*n+3)) / (n^3 * (2*n-1) * C(2*n,n) * C(4*n,2*n)^2) (from the Lupas 2000 reference).
G = phi(-1, 2, 1/2)/4 = A247685/4, where phi is Lerch transcendent. - Jean-François Alcover, Mar 28 2013
G = (1/2)*Integral_{x=0..Pi/2} log(cot(x)+csc(x)) dx. - Jean-François Alcover, Apr 11 2013 [see the Adamchik link]
G = -Integral_{x=0..1} (log x)/(1+x^2) dx = Integral_{x>=1} (log x)/(1+x^2) dx. - Clark Kimberling, Nov 04 2016
G = (Zeta(2, 1/4) - Pi^2)/8 = (Psi(1, 1/4) - Pi^2)/8 = (A282823-Pi^2)/8, with the Hurwitz zeta function and the trigamma function Psi(1, z). For the partial sums of the series given in the name see A294970/A294971. - Wolfdieter Lang, Nov 15 2017
Equals Im(Li_{2}(i)). - Peter Luschny, Oct 04 2019
Equals -Integral_{x=0..Pi/4} log(tan(x)) dx. - Amiram Eldar, Jun 29 2020
Equals (1/2)*Integral_{x=0..1} K(x) dx = -1/2 + Integral_{x=0..1} E(x) dx, where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively, as a functions of the elliptic modulus k. - Gleb Koloskov, Jun 25 2021
From Peter Bala, Dec 08 2021: (Start)
G = 1/2 + 4*Sum_{n >= 1} (-1)^(n+1)*n/(4*n^2 - 1)^2 = -13/18 + (2^7)*3*Sum_{n >= 1} (-1)^(n+1)*n/((4*n^2 - 1)^2*(4*n^2 - 9)^2) = -3983/1350 + (2^15)*3*5*Sum_{n >= 1} (-1)^(n+1)*n/((4*n^2 - 1)^2*(4*n^2 - 9)^2*(4*n^2 - 25)^2).
G = 3/2 - 16*Sum_{n >= 1} (-1)^(n+1)*n/(4*n^2 - 1)^3 = 401/6 - (2^13)*(3^3)*Sum_{n >= 1} (-1)^n*n/((4*n^2 - 1)^3*(4*n^2 - 9)^3) = 5255281/1350 - (2^25)*(3^3)*(5^3)*Sum_{n >= 1} (-1)^(n+1)*n/((4*n^2 - 1)^3*(4*n^2 - 9)^3*(4*n^2 - 25)^3). (End)
From Amiram Eldar, Jan 07 2024: (Start)
Equals beta(2), where beta is the Dirichlet beta function.
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^2)^(-1). (End)
Equals 2*Integral_{x=0..Pi/4} log(2*cos(x)) dx = -2*Integral_{x=0..Pi/4} log(2*sin(x)) dx (see Finch). - Stefano Spezia, Nov 14 2024
Equals Integral_{x=0..Pi/4} log((1 + tan(x))/(1 - tan(x))) dx. - Kritsada Moomuang, Jun 03 2025

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), May 28 2002

A294970 Numerators of the partial sums for the Catalan constant A006752: Sum_{k=0..n} ((-1)^k)/(2*k+1)^2, n >= 0.

Original entry on oeis.org

1, 8, 209, 10016, 91369, 10956424, 1863641881, 1854623872, 538015351033, 193637145687688, 194117166024913, 102476291858462752, 2566386635039604121, 23062916917686411464, 19421109407275720721849, 18642496069331249273291264
Offset: 0

Views

Author

Wolfdieter Lang, Nov 15 2017

Keywords

Comments

The corresponding denominators are given in A294971.

Examples

			The rationals r(n) begin: 1, 8/9, 209/225, 10016/11025, 91369/99225, 10956424/12006225, 1863641881/2029052025, 1854623872/2029052025, 538015351033/586396035225, 193637145687688/211688968716225, 194117166024913/211688968716225, 102476291858462752/111983464450883025, ...
r(10^5) = 0.9159655942 (Maple 10 digits) to be compares with 0.91596559417... from A006752.

Links

Crossrefs

Cf. A006752, A294971.

Programs

  • Magma
    [Numerator((&+[(-1)^k/(2*k+1)^2: k in [0..n]])): n in [0..20]]; // G. C. Greubel, Aug 22 2018
  • Mathematica
    Table[Numerator[Sum[(-1)^k/(2*k+1)^2, {k,0,n}]], {n,0,20}] (* Vaclav Kotesovec, Nov 15 2017 *)
  • PARI
    for(n=0,20, print1(numerator(sum(k=0,n, (-1)^k/(2*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018

Formula

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k/(2*k+1)^2 = (Zeta(2, 1/4) - Zeta(2,floor(n/2) + 5/4) - (Zeta(2, 3/4) - Zeta(2,floor((n-1)/2) + 7/4)))/16, Zeta(2, z) = Psi(1, z), with the Hurwitz Zeta function and the trigamma function Psi(1, z).
The limit n-> infinity of r(n) is the Catalan constant given in A006752; see in particular the formula (Zeta(2, 1/4) - Zeta(2, 3/4))/16.

A319150 a(n) = gcd(A275286(n), A001818(n+1)) / A025549(n+1)^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 19, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1
Offset: 0

Views

Author

Tristan Cam, Nov 08 2018

Keywords

Comments

A(n) = a(n)*A025549(n+1)^2 = gcd(B(n), C(n)).
B(n) = A275286(n).
C(n) = A001818(n+1).
At first gcd(B(n), C(n)) = A025549(n+1)^2, but from n = 27 to n = 37, gcd(B(n), C(n)) = 11*A025549(n+1)^2, and then comes back to normal, then equals 19*A025549(n+1)^2, comes back to normal again, and so on ...
Let S(n) = Sum_{k=0..n} ((-1)^k)/(2*k+1)^2 (S(n) is NOT an integer sequence).
Notice that when n approaches +oo, D(n) converges to Catalan's constant (A006752).
A294970(n) is equal to the numerator of S(n) (when reduced).
Therefore B(n)/A(n) = A294970(n)
A294971(n) is equal to the denominator of S(n) (when reduced).
Therefore C(n)/A(n) = A294971(n).
This sequence was used to study the expression B(n)/C(n) (which equals S(n)) in an attempt to find out if Catalan's constant is irrational.

Examples

			For n = 5:
B(n) = A275286(5) = 98607816;
C(n) = A001818(5+1) = 108056025;
gcd(98607816,108056025) = A(5) = 9;
A025549(5+1)^2 = 3^2 = 9;
So a(5) = A(5)/A025549(5+1)^2 = 9/9 = 1.

Crossrefs

Cf. A275286, A001818, A025549, A294970, A294971, A006752.

Programs

  • Mathematica
    a[n_] := GCD[(2n+1)!!^2 * Sum[(-1)^k/(2k+1)^2, {k, 0, n}], (2n+1)!!^2]*
LCM @@ Range[1, 2n+1, 2]^2 / ((2n+1)!!)^2; Array[a, 100, 0] (* Amiram Eldar, Nov 16 2018 *)
  • PARI
    dfo(n) = (2*n)! / n! / 2^n;
a6(n) = dfo(n+1)^2*sum(k=0, n, (-1)^k/(2*k+1)^2);
a8(n) = ((2*n)!/(n!*2^n))^2;
a9(n) = (((2*n)!/n!)/2^n)/lcm(vector(n, i, 2*i-1));
a(n) = gcd(a6(n) , a8(n+1)) / a9(n+1)^2; \\ Michel Marcus, Nov 08 2018

Formula

Explicit formula:
a(n) = gcd( ((2*n+1)!!)^2 * (Sum_{i=0..n}((-1)^i)/(2*i+1)^2), ((2*n+1)!!)^2 ) / ( (((2*n+1)!!)^2) / ( lcm{1,3,5,...,2*n+1} ) )^2.
A few relations:
gcd(A275286(n), A001818(n+1)) = a(n)*A025549(n+1)^2 = A(n);
A275286(n)/A(n) = A294970(n);
A001818(n+1)/A(n) = A294971(n);
Limit_{n->+oo} A294970(n)/A294971(n) = G (Catalan's Constant, decimal expansion: A006752).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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