A088538 -id:A088538 - cp's OEIS frontend

A132701 Decimal expansion of 11/Pi.

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

Offset: 1

Omar E. Pol, Aug 26 2007

			3.501408748021697...

Index entries for transcendental numbers

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

Digits:=100; evalf(11/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[11/Pi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2009 *)

11/Pi \\ Charles R Greathouse IV, Dec 31 2011

More terms from Vladimir Joseph Stephan Orlovsky, Jun 19 2009

A218387 Decimal expansion of the spanning tree constant of the square lattice.

Original entry on oeis.org

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

Offset: 1

R. J. Mathar, Oct 27 2012

			1.16624361612327512055353782587357967545626461594...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 1.7 and 5.22.6, pp. 54, 399.
Asmus L. Schmidt, Ergodic theory of complex continued fractions, Number Theory with an Emphasis on the Markoff Spectrum, in: A. D. Pollington and W. Moran (eds.), Number Theory with an Emphasis on the Markoff Spectrum, Dekker, 1993, pp. 215-226.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Anthony J. Guttmann, Spanning tree generating functions and Mahler measure, arXiv:1207.2815 [math-ph], 2012.
Sheldon Yang, Some properties of Catalan's constant G, Internat. J. Math. Ed. Sci. Tech. 23 (4) (1992) 549-556, L*(1).

Cf. A006752 (Catalan), A088538 (4/Pi), A229728, A247685.

R:= RealField(100); 4*Catalan(R)/Pi(R); // G. C. Greubel, Aug 23 2018

Maple
```
evalf(Catalan*4/Pi) ;
```

RealDigits[4*Catalan/Pi, 10, 100][[1]] (* G. C. Greubel, Aug 23 2018 *)

default(realprecision, 100); 4*Catalan/Pi \\ G. C. Greubel, Aug 23 2018

Equals the product of A006752 by A088538.
From Amiram Eldar, Jul 22 2020: (Start)
Equals 1 + Sum_{k>=1} (2*k-1)!!^2/((2*k)!!^2 * (2*k + 1)).
Equals Sum_{k>=0} binomial(2*k,k)^2/(16^k * (2*k + 1)). (End)
Equals (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)^2) / (Sum_{n>=1} (-1)^(n+1)/(2*n - 1)) [Schmidt] (see Finch). - Stefano Spezia, Nov 07 2024
Equals log(A229728) = A247685/Pi. - Hugo Pfoertner, Nov 07 2024
Equals Integral_{x=0..1} EllipticK(x)/(Pi*sqrt(x)) dx. - Kritsada Moomuang, Jun 21 2025

A076342 a(n) = A076340(A000040(n)), real part of primes mapped as defined in A076340, A076341.

Original entry on oeis.org

2, 4, 4, 8, 12, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 60, 60, 68, 72, 72, 80, 84, 88, 96, 100, 104, 108, 108, 112, 128, 132, 136, 140, 148, 152, 156, 164, 168, 172, 180, 180, 192, 192, 196, 200, 212, 224, 228, 228, 232, 240, 240, 252, 256, 264, 268, 272

Offset: 1

Reinhard Zumkeller, Oct 08 2002

By definition of the map defined in A076340, A076341: 2->(2,0) and p->((floor(p/4)+floor((p mod 4)/2))*4,2-(p mod 4)) for odd primes p.

Number of solutions to x^2 + y^2 = 1 (mod p). - Lekraj Beedassy, Oct 22 2004

			A000040(11)=31=(32-1) -> (32,-1), therefore a(11)=32 and A070750(11)=-1.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040, A070750, A076340, A076341, A082542, A088538.

f:= proc(n) local p;
  p:= ithprime(n);
  if p mod 4 = 1 then p-1 elif p mod 4 = 3 then p+1 else 2 fi
end proc:
map(f, [$1..100]); # Robert Israel, Dec 26 2016

a[1] = 2; a[n_] := With[{p = Prime[n]}, p - JacobiSymbol[-1, p]]; Array[a, 60] (* Jean-François Alcover, Feb 01 2018, after Lekraj Beedassy *)
a[n_] := Prime[n] - 2 + Mod[Prime[n], 4]; Array[a, 100] (* Amiram Eldar, Dec 24 2022 *)

a(n) = p-(-1/p) = p+(-1)^{(p+1)/2} for an odd prime p. {(a/b) stands for the value of the Legendre symbol}. - Lekraj Beedassy, Oct 22 2004
From Amiram Eldar, Dec 24 2022: (Start)
a(n) = A000040(n) - A070750(n).
a(n) = A100484(n) - A082542(n).
Product_{n>=1} a(n)/prime(n) = 4/Pi (A088538). (End)

A211074 Decimal expansion of 4/Pi - 1/2.

Original entry on oeis.org

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

Offset: 0

Charles R Greathouse IV, May 02 2012

Equivalent resistance between two nodes on an infinite rectangular lattice of ideal unit resistors, where the nodes are separated by two resistors along one axis and one resistor on the other.

			0.77323954473516268615107010698011489627567716592365158998133875247117...

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Author?, Infinite 2D square grid of 1-ohm resistors, April 30 2004.
D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Aug 14 1998.
D. Atkinson and F. J. van Steenwijk, Infinite resistive lattices, Am. J. Phys. 67 (1999), 486-492.
Matthew Beckler, Infinite Grid of Resistors - Solution by Simulation, Dec 16 2009.
Kevin Brown, Infinite Grid of Resistors.
J. Cserti, Application of the lattice Green's function for calculating the resistance of an infinite networks of resistors, arXiv:cond-mat/9909120 [cond-mat.mes-hall], 1999-2000.
Peter G. Doyle and J. Laurie Snell, Random walks and electric networks (2006), section 1.1.4.
Alan Eustace, Pencils down, people (official Google Blog article, archived on archive.org), Sep 30 2004.
Randall Munroe, Nerd Sniping, xkcd Web Comic #356, Dec 12 2007.
Michael Pusateri, Google Labs Aptitude Test, Sep 13 2004.
Index entries for transcendental numbers

Cf. A088538, A355565.

RealDigits[4/Pi - 1/2, 10, 87][[1]] (* modified by Harvey P. Dale, Jan 14 2015 *)

PARI
```
4/Pi-1/2
```

A120669 Decimal expansion of arccos(1-8/(Pi^2)).

Original entry on oeis.org

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

Offset: 1

Rick L. Shepherd, Jun 22 2006

For a circle with radius r, the measurement in radians of the central angle with endpoints on the circle that are r*4/Pi apart: The average central angle (<= Pi) formed using two randomly chosen points on a circle. The average arc length between such endpoints is r*A120669 corresponding to the average chord length r*A088538; so for the unit circle arc length is A120669 and chord length is A088538.

			1.38021418274907990400875581814...

Cf. A088538, A120670 (same in degrees), A120671 (A120669/2Pi).

RealDigits[ArcCos[1-8/Pi^2],10,120][[1]] (* Harvey P. Dale, Dec 15 2014 *)

PARI
```
acos(1-8/Pi^2)
```

A120670 Decimal expansion of 180*arccos(1-8/(Pi^2))/Pi.

Original entry on oeis.org

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

Offset: 2

Rick L. Shepherd, Jun 22 2006

For a circle with radius r, the measurement in degrees of the central angle with endpoints on the circle that are r*4/Pi apart: The average central angle (<= 180 degrees) formed using two randomly chosen points on a circle. The average arc length between such endpoints is r*A120669 corresponding to the average chord length r*A088538; so for the unit circle arc length is A120669 and chord length is A088538.

			79.0804474956203908253980311180...

Cf. A088538, A120669 (same in radians), A120671 (A120670/360).

RealDigits[180 ArcCos[1-8/Pi^2]/Pi,10,120][[1]] (* Harvey P. Dale, Feb 17 2023 *)

PARI
```
180*acos(1-8/Pi^2)/Pi
```

Equals 180*A120669/Pi = 360*A120671.

A120671 Decimal expansion of arccos(1-8/(Pi^2)) / (2*Pi).

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jun 22 2006

The average arc length between two randomly chosen points on a circle of radius r is r*A120669 corresponding to the average chord length r*A088538. (Each arc averaged is no larger than the semicircle.). This sequence is the ratio of that average arc length to the circumference (and is thus also the ratio of the corresponding sector's area to the circle's area).

			0.219667909710056641181661197550...

Cf. A088538, A120669 (2*Pi*A120671), A120670 (360*A120671).

RealDigits[ArcCos[(1-(8/(Pi^2) ))]/(2Pi),10,120][[1]] (* Harvey P. Dale, Jun 17 2018 *)

PARI
```
acos(1-8/Pi^2)/(2*Pi)
```

Equals A120669/(2*Pi) = A120670/360.

A132698 Decimal expansion of 8/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

			2.5464790894703253723021402139602297925513543318473031799626775049423487621476....

Index entries for transcendental numbers

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

Digits:=100; evalf(8/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[8/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

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

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

A132714 Decimal expansion of 24/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			=7.639437268410976116906420641880689377654062995541909539888032514827...

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for transcendental numbers

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

RealDigits[N[24/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 03 2009 *)

PARI
```
24/Pi \\ G. C. Greubel, Feb 20 2017
```

24/Pi = Sum_{k>=0} ( (30*k+7)*C(2*k,k)^2*(Hypergeometric2F1[1/2 - k/2, -k/2, 1, 64])/(-256)^k ). - Alexander R. Povolotsky, Dec 20 2012

Another version of this identity is: Sum[(30*k+7) * Binomial[2k,k]^2 * (Sum[Binomial[k-m,m] * Binomial[k,m] * 16^m, {m,0,k/2}])/(256)^k, {k,0,infinity}]. - Alexander R. Povolotsky, Jan 25 2013

More terms from Vladimir Joseph Stephan Orlovsky, Dec 03 2009

A277235 Decimal expansion of 2/(Gamma(3/4))^4.

Original entry on oeis.org

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

Offset: 0

Wolfdieter Lang, Nov 13 2016

This is the value of one of Ramanujan's series: 1 - 5*(1/2)^5 + 9*(1*3/(2*4))^5 -13*(1*3*5/(2*4*6))^5 + - ... . See the Hardy reference p.7. eq. (1.4) and pp. 105-106. For the partial sums see A278140.

The proof of Hardy and Whipple mentioned in the Hardy reference reduces this series to (2/Pi)*Morley's series (for m=1/2). For this series see A277232 and A091670.

			2/Gamma(3/4)^4 = 0.88694116857811540541152...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publ., Providence, RI, 2002, pp. 7, 105-106, 111.

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

Cf. A014549, A060294, A074799, A074800, A088538, A091670, A263809, A277232, A278140.

SetDefaultRealField(RealField(100)); 2/(Gamma(3/4))^4; // G. C. Greubel, Oct 26 2018

RealDigits[2/(Gamma[3/4])^4, 10, 100][[1]] (* G. C. Greubel, Oct 26 2018 *)

2/gamma(3/4)^4 \\ Michel Marcus, Nov 13 2016

Equals Sum_{k=0..n} (1+4*k)*(binomial(-1/2,k))^5 = Sum_{k=0..n} (-1)^k*(1+4*k)*((2*k-1)!!/(2*k)!!)^5. The double factorials are given in A001147 and A000165 with (-1)!! := 1.
Equals A060294 * A091670.
For (1+4*k)*((2*k-1)!!/(2*k)!!)^5 see A074799(k) / A074800(k).
From Amiram Eldar, Jul 13 2023: (Start)
Equals (Gamma(1/4)/Pi)^4/2.
Equals A088538 * A014549^2.
Equals A263809/Pi. (End)

cp's OEIS Frontend

A132701 Decimal expansion of 11/Pi.

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A218387 Decimal expansion of the spanning tree constant of the square lattice.

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A076342 a(n) = A076340(A000040(n)), real part of primes mapped as defined in A076340, A076341.

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A211074 Decimal expansion of 4/Pi - 1/2.

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Mathematica

PARI

A120669 Decimal expansion of arccos(1-8/(Pi^2)).

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Mathematica

PARI

A120670 Decimal expansion of 180*arccos(1-8/(Pi^2))/Pi.

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Mathematica

PARI

Formula

A120671 Decimal expansion of arccos(1-8/(Pi^2)) / (2*Pi).

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Mathematica