A006752 -id:A006752 - cp's OEIS frontend

A104338 Binary expansion of Catalan constant A006752.

1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane, Sep 16 2007

			In base 10: 0.91596559..., in base 2: 0.1110101001111...

G. C. Greubel, Table of n, a(n) for n = 0..10000

Cf. A006752, A113860.

Cf. A006752, A014538, A153069, A153070, A054543, A118323. - Stuart Clary, Dec 17 2008

RealDigits[Catalan, 2, 200] (* Stefan Steinerberger, Sep 17 2007 *)

More terms from Stefan Steinerberger, Sep 17 2007

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

Wolfdieter Lang, Nov 15 2017

The corresponding denominators are given in A294971.

			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.

G. C. Greubel, Table of n, a(n) for n = 0..575
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function
Eric Weisstein's World of Mathematics, Trigamma Function

Cf. A006752, A294971.

[Numerator((&+[(-1)^k/(2*k+1)^2: k in [0..n]])): n in [0..20]]; // G. C. Greubel, Aug 22 2018

Table[Numerator[Sum[(-1)^k/(2*k+1)^2, {k,0,n}]], {n,0,20}] (* Vaclav Kotesovec, Nov 15 2017 *)

for(n=0,20, print1(numerator(sum(k=0,n, (-1)^k/(2*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018

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.

A294971 Denominators 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, 9, 225, 11025, 99225, 12006225, 2029052025, 2029052025, 586396035225, 211688968716225, 211688968716225, 111983464450883025, 2799586611272075625, 25196279501448680625, 21190071060718340405625, 20363658289350325129805625

Offset: 0

Wolfdieter Lang, Nov 15 2017

The corresponding numerators are given in A294970. There details are given.

			See A294970.

G. C. Greubel, Table of n, a(n) for n = 0..575

Cf. A006752, A294970.

[Denominator((&+[(-1)^k/(2*k+1)^2: k in [0..n]])): n in [0..20]]; // G. C. Greubel, Aug 22 2018

Table[Denominator[Sum[(-1)^k/(2*k+1)^2, {k,0,n}]], {n,0,20}] (* Vaclav Kotesovec, Nov 15 2017 *)

for(n=0,20, print1(denominator(sum(k=0,n, (-1)^k/(2*k+1)^2)), ", ")) \\ G. C. Greubel, Aug 22 2018

a(n) = numerator(r(n)) with the rationals r(n) = Sum_{k=0..n} (-1)^k/(2*k+1)^2.

For r(n) in terms of the Hurwitz Zeta function or the trigamma function see A294970.

A099616 Sum of the first n decimal places of Catalan's constant 0.9159655941772... (sequence A006752).

Original entry on oeis.org

9, 10, 15, 24, 30, 35, 40, 49, 53, 54, 61, 68, 70, 71, 80, 80, 81, 86, 86, 91, 95, 101, 101, 104, 109, 110, 114, 123, 126, 128, 131, 139, 143, 144, 145, 145, 152, 159, 163, 164, 168, 177, 180, 187, 191, 193, 201, 202, 208, 215, 217, 218, 221, 225, 227, 233, 239

Offset: 1

Mark Hudson (mrmarkhudson(AT)hotmail.com), Oct 25 2004

			Catalan's constant = 0.9159655941772... so the sums are 9, 9+1, 9+1+5, 9+1+5+9, 9+1+5+9+6..., leading to the terms 9, 10, 15, 24, 30, ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A006752 for digits of Catalan's constant. Similarly constructed sequences for other constants at A099541 and sequence references therein.

Accumulate[RealDigits[N[Catalan,60]][[1]]] (* Harvey P. Dale, May 10 2011 *)

A113860 Start with the binary representation of the Catalan constant (A104338, A006752) = 0.91596559... = sum_{i=1..infinity} b(i)/2^i, where b(i)=1,1,1,0,1,0,1,0,0,1,1,1,1.... Then a(n-1)=sum_{i=1..k: sum_{ j = 1..k} b(j)=n} b(i) * 2^(i-1). In words: scan the binary digits of the number, halt at each nonzero binary digit, add a power of 2 corresponding to the place of this digit after the comma, assign current partial sum to a(n), increment n.

Original entry on oeis.org

1, 3, 7, 23, 87, 599, 1623, 3671, 7767, 15959, 81495, 343639, 867927, 1916503, 18693719, 152911447, 421346903, 958217815, 2031959639, 4179443287, 12769377879, 1112281005655, 9908374027863, 27500560072279, 97869304249943

Offset: 0

Artur Jasinski, Jan 25 2006

An instance of a Jasinski Integer Sequence using the convention JIS[number,counting system] as defined for example in A080355. This is JIS [Catalan constant,binary]=JIS[0.9159655941772190150546..,2].

Cf. A080355, A080567, A099969, A099970, A099971, A099972.

Naming a sequence after oneself is deprecated. - N. J. A. Sloane.

Corrected and extended by R. J. Mathar, Aug 31 2007

A345738 Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 25 2021

A projectile is launched with an initial speed v at angle theta above the horizon. Assuming that the gravitational acceleration g is uniform and neglecting the air resistance, the trajectory is a part of a parabola whose expected length, averaged over theta uniformly chosen at random from the range [0, Pi/2], is c * v^2/g, where c is this constant.

The length of the trajectory as a function of theta is L(theta) = (v^2/g)*(sin(theta) + cos(theta)^2*log((1+sin(theta))/(1-sin(theta)))/2). L(theta) goes from 0 to 1 between theta = 0 and Pi/2. It has a maximum at theta = 0.985514... (A345737), and a unique value at 0 <= theta < 0.599677... (A345739). The average length (c * v^2/g) occurs at theta = 0.5152731296... (29.522975... degrees).

			0.90143169424542823181453643968181856179705159945258...

Péter Kórus, Notes on Projectile Motion, The American Mathematical Monthly, Vol. 126, No. 4 (2019), pp. 358-360.

Cf. A000796, A006752, A143233, A345737, A345739.

RealDigits[(2*Catalan + 1)/Pi, 10, 100][[1]]

Equals (2 * A006752 + 1)/A000796.

Equals 2 * A143233 + 1.

A377753 Decimal expansion of 8*G/Pi^2, where G is the Catalan's constant (A006752).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Nov 07 2024

			0.7424537454215443259079279607988799424377218365...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 7.7, pp. 54, 474.

Cf. A002388, A006752, A218387, A242822 (see the second formula).

Mathematica
```
RealDigits[8Catalan/Pi^2,10,100][[1]]
```

8*Catalan/Pi^2 \\ Charles R Greathouse IV, Nov 27 2024

Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} 1/(2*n - 1)^2) (see Finch).
Equals 1/A242822. - Hugo Pfoertner, Nov 07 2024
Equals (1 - W)/(1 + W), where W = tanh(Sum_{prime p == 3 (mod 4)} arctanh(1/p^2)) = zeta(2,3/4)/zeta(2,1/4) = (Pi^2 - 8*G)/(Pi^2 + 8*G) = 0.1478066521164... Physical interpretation: the constant W is the relativistic sum of the velocities c/p^2 over all primes p == 3 (mod 4), in units where the speed of light c = 1. - Thomas Ordowski, Nov 23 2024

A132201 Pierce expansion of Catalan's Constant A006752.

Original entry on oeis.org

1, 11, 13, 59, 582, 12285, 127893, 654577, 1896651, 2083263, 3828867, 6195679, 22339606, 43877386, 209882043, 269091773, 1585394894, 2614512078, 3726537414, 4487682121, 6296491774, 8648456991, 23933983277, 174313954158, 367633382556

Offset: 1

R. J. Mathar, Nov 05 2007

nonn

			0.9159... = 1/1 - 1/11 + 1/(11*13) - 1/(11*13*59) + 1/(11*13*59*582) - ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Am. Math. Monthly 36 (10) (1929) 523.
Eric Weisstein's World of Mathematics, Pierce Expansion.

Cf. A006752.

Digits := 300: Pierce := proc(x) local resid,a,i,an ; resid := x ; a := [] ; for i from 1 do an := floor(1./resid) ; a := [op(a),an] ; resid := evalf(1.-an*resid) ; if ilog10( mul(i,i=a)) > 0.7*Digits then break ; fi ; od: RETURN(a) ; end: Pierce(Catalan);

PierceExp[A_, n_] := Join[Array[1 &, Floor[A]], First@Transpose@ NestList[{Floor[1/Expand[1 - #[[1]] #[[2]]]], Expand[1 - #[[1]] #[[2]]]} &, {Floor[1/(A - Floor[A])], A - Floor[A]}, n - 1]]; PierceExp[N[Catalan , 7!], 20] (* G. C. Greubel, Nov 15 2016 *)

r=1/Catalan; for(n=1, 10, print(floor(r), ", "); r=r/(r-floor(r))) \\ G. C. Greubel, Nov 15 2016

A322757 Decimal expansion of G/(2*Pi), where G is Catalan's constant A006752.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Dec 28 2018

Per-site entropy of the dimer model on a square grid.

			0.1457804520154093900691922282341974594320330769929186351305007845558738184...

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 554.

Cf. A006752, A019692, A143233.

RealDigits[Catalan/(2*Pi), 10, 120][[1]] (* Amiram Eldar, May 24 2023 *)

Equals (1/2) * A143233. - Peter Bala, Aug 01 2025

A367053 Decimal expansion of Catalan's constant minus Serret's integral, A006752 - A102886.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Nov 03 2023

			0.64376733288926874874201740265268236735682641173551134747577...

Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Wikipedia, Bailey-Borwein-Plouffe formula.

Cf. A006752, A102886.

Im(polylog(2, (1 + I)/2)): evalf(%, 88);

First[RealDigits[Catalan - Pi * Log[2] / 8, 10, 87]]

# Use a few guard digits when computing.
# BBP formula (1 / 16) P(2, 16, 8, (8, 8, 4, 0, -2, -2, -1, 0))
from decimal import Decimal as dec, getcontext
def BBPCatSer(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(16)
    for k in range(n):
        ek = dec(8 * k)
        s += f * ( dec(8) / (ek + 1) ** 2 + dec(8) / (ek + 2) ** 2
                 + dec(4) / (ek + 3) ** 2 - dec(2) / (ek + 5) ** 2
                 - dec(2) / (ek + 6) ** 2 - dec(1) / (ek + 7) ** 2 )
        f /= g
    return s / g
print(BBPCatSer(200))

Equals Integral_{x=0..1} arctan(x)/(x*(1 + x)) dx.
Equals Im(Polylog(2, (1 + i)/2)).
Equals Catalan - Pi * log(2) / 8.
Equals (zeta(2, 1/4) - Pi * (Pi + log(2))) / 8.

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A104338 Binary expansion of Catalan constant A006752.

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

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A294971 Denominators of the partial sums for the Catalan constant A006752: Sum_{k=0..n} ((-1)^k)/(2*k+1)^2, n >= 0.

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A099616 Sum of the first n decimal places of Catalan's constant 0.9159655941772... (sequence A006752).

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A345738 Decimal expansion of (2*G+1)/Pi, where G is Catalan's constant (A006752).

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A377753 Decimal expansion of 8*G/Pi^2, where G is the Catalan's constant (A006752).

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A132201 Pierce expansion of Catalan's Constant A006752.

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