A086201 -id:A086201 - cp's OEIS frontend

A132697 Decimal expansion of 7/Pi.

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

Offset: 1

Omar E. Pol, Aug 26 2007

			2.22816920328653470076437268721520106848243504036639028246734281682455516687917....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201.

Digits:=100; evalf(7/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[7/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

7/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132721 Decimal expansion of 31/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			9.867606471697510817670793...

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696, A132697, A132698, A132699, A132701, A132702.

RealDigits[31/Pi,10,120][[1]] (* Harvey P. Dale, Dec 28 2021 *)

31/Pi \\ Charles R Greathouse IV, Oct 01 2022

Terms a(32) and beyond from Andrew Howroyd, Jan 03 2020

A233527 Decimal expansion of arctan( 1/(2*Pi) ): opposite angle for a right triangle of equal area to a circle.

Original entry on oeis.org

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

Offset: 0

John W. Nicholson, Dec 11 2013

			0.15783119028815886016760874178281892360507124059905106041555796721382...

Wikipedia, Area of a circle: Triangle proof

Cf. A019669, A019692, A086201, A233528.

RealDigits[ArcTan[1/(2 Pi)], 10, 110][[1]] (* Bruno Berselli, Dec 16 2013 *)

PARI
```
atan(1/(2*Pi))
```

Equals arctan(1/A019692).

Equals A019669 - A233528. [Bruno Berselli, Dec 16 2013]

More terms from Ralf Stephan, Dec 16 2013

A110191 Decimal expansion of 1/6 - 1/(2*Pi).

Original entry on oeis.org

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

Offset: 0

Eric W. Weisstein, Jul 15 2005

			0.007511723574771330897...

A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol. 1 (Overseas Publishers Association, Amsterdam, 1986), p. 722, section 5.3.5, formula 9.

Bruce C. Berndt and K. Venkatachaliengar, On the transformation formula for the Dedekind eta-function, in: F. G. Garvan and M. E. H. Ismail (eds.), Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics, eds., Kluwer, Dordrecht, 2001, pp. 73-77; preprint.
Mark W. Coffey, Bernoulli identities, zeta relations, determinant expressions, Mellin transforms, and representation of the Hurwitz numbers, Journal of Number Theory, Vol. 184 (2018), pp. 27-67, see Lemma 2, p. 62; arXiv preprint, arXiv:1601.01673 [math.NT], 2016, see Lemma 2, p. 33.
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant.
Index entries for transcendental numbers.

Cf. A086201.

RealDigits[1/6 - 1/(2*Pi), 10, 120, -1][[1]] (* Amiram Eldar, Jun 15 2023 *)

-1/(2*Pi) + 1/6 \\ Michel Marcus, Jan 11 2016

-suminf(k=1, 1/sin(k*Pi*I)^2) \\ Michel Marcus, Jan 11 2016

suminf(k=1, 1/sinh(k*Pi)^2) \\ Vaclav Kotesovec, May 19 2022

Equals -Sum_{k>=1} 1/sin(k*Pi*i)^2. - Michel Marcus, Jan 11 2016

Equals Sum_{k>=1} 1/sinh(k*Pi)^2. - Vaclav Kotesovec, May 19 2022

A132717 Decimal expansion of 27/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			8.59436692696234813151972322211577554986082086998464...

Index entries for transcendental numbers.

Cf. A049541, A060294, A089491, A088538, A086201, A132696, A132697, A132698, A132699, A132701, A132702.

RealDigits[27/Pi, 10, 120][[1]] (* Amiram Eldar, Jun 13 2023 *)

27/Pi \\ Charles R Greathouse IV, Oct 01 2022

A218441 a(n) = A000108(n)*A001764(n).

Original entry on oeis.org

1, 1, 6, 60, 770, 11466, 188496, 3325608, 61866090, 1199333850, 24030289140, 494663027040, 10414559269296, 223487031938800, 4874879691748800, 107852781825352080, 2415945569351185530, 54714061423541554650, 1251237165698155135500, 28864572348777684057000

Offset: 0

Paul D. Hanna, Oct 28 2012

G.f. of A000108, C(x), satisfies: C(x) = 1 + x*C(x)^2;

G.f. of A001764, F(x), satisfies: F(x) = 1 + x*F(x)^3.

			G.f.: A(x) = 1 + x + 6*x^2 + 60*x^3 + 770*x^4 + 11466*x^5 + 188496*x^6 + ...

Amiram Eldar, Table of n, a(n) for n = 0..705

Cf. A000108, A001764, A086201.

a[n_] := CatalanNumber[n] * Binomial[3*n, n]/(2*n+1); Array[a, 20, 0] (* Amiram Eldar, Apr 26 2025 *)

A218441[n]:=binomial(2*n, n)/(n+1)*binomial(3*n, n)/(2*n+1)$
makelist(A218441[n],n,0,30); /* Martin Ettl, Oct 29 2012 */

{a(n)=binomial(2*n,n)/(n+1)*binomial(3*n,n)/(2*n+1)}
for(n=0,25,print1(a(n),", "))

a(n) ~ 3^(3*n+1/2)/(2*Pi*n*(n+1)*(2*n+1)) = A086201*3^(3*n+1/2)/(n*(n+1)*(2*n+1)) (using the Stirling approximation for n!). - A.H.M. Smeets, Dec 31 2022

A132703 Decimal expansion of 13/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			4.138028520389278729990977847685373412895950789251867667439350945531316738....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696-A132699, A132701, A132702.

Digits:=100; evalf(13/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[13/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

13/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132704 Decimal expansion of 14/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			4.45633840657306940152874537443040213696487008073278056493468563364911033....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696-A132699, A132701, A132702.

Digits:=100; evalf(14/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[14/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

14/Pi \\ Charles R Greathouse IV, Oct 01 2022

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

More terms from Harvey P. Dale, Oct 03 2012

A132706 Decimal expansion of 16/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			5.092958178940650744604280427920459585102708663694606359925355....

Bruce C. Berndt, Ramanujan’s Notebooks, Part II, Springer-Verlag, New York, 1989.

Index entries for transcendental numbers

Cf. A019683, A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

RealDigits[N[16/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

16/Pi \\ Charles R Greathouse IV, Oct 01 2022

Equals 4 + Sum_{k>=0} binomial(2*k,k)^2/((k+1)^2*16^k). - Amiram Eldar, May 21 2021

16/Pi = 5 + 1^2/(10 + 3^2/(10 + 5^2/(10 + ...))). See Berndt, Entry 25, p. 140, with n = 0 and x = 5. - Peter Bala, Feb 18 2024

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132707 Decimal expansion of 17/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			5.4112680651244414161420479546654883091716279551755192574206896980024911195637....

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A132696 to A132699, A132701, A132702.

RealDigits[N[17/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

17/Pi \\ Charles R Greathouse IV, Oct 01 2022

cp's OEIS Frontend

A132697 Decimal expansion of 7/Pi.

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Maple

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A132721 Decimal expansion of 31/Pi.

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A233527 Decimal expansion of arctan( 1/(2*Pi) ): opposite angle for a right triangle of equal area to a circle.

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A110191 Decimal expansion of 1/6 - 1/(2*Pi).

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A132717 Decimal expansion of 27/Pi.

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A218441 a(n) = A000108(n)*A001764(n).

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Maxima

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A132703 Decimal expansion of 13/Pi.

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