A089491 -id:A089491 - cp's OEIS frontend

A132721 Decimal expansion of 31/Pi.

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

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

A268085 a(n) = Catalan(n)^2*n.

Original entry on oeis.org

0, 1, 8, 75, 784, 8820, 104544, 1288287, 16359200, 212751396, 2821056160, 38013731756, 519227905728, 7174705330000, 100136810390400, 1409850293610375, 20002637245262400, 285732116760449700, 4106497099278420000, 59341164471850545900, 861753537765219528000

Offset: 0

Ralf Steiner, Jan 26 2016

The series whose terms are the quotients a(n)/A013709(n) is convergent to 1-3/Pi.(see formula).
Proof: Both the Wallis-Lambert-series-1=4/Pi-1 and the elliptic Euler-series=1-2/Pi are absolutely convergent series. Thus any linear combination of the terms of these series will be also absolutely convergent to the value of the linear combination of these series - in this case to 1-3/Pi. Q.E.D.
Apart from inclusion of a(0) the same as A145600. - R. J. Mathar, Feb 07 2016

			For n=3 the a(3)= 75.

Ralf Steiner, Beispiele zur modifizierten Wallis-Lambert-Reihe (in German).

Cf. A000108, A013709.

[Catalan(n)^2*n: n in [0..20]]; // Vincenzo Librandi, Jan 26 2016

Mathematica
```
Table[CatalanNumber[n]^2 n, {n, 0, 20}]
```

a(n) = n*(binomial(2*n, n)/(n+1))^2; \\ Altug Alkan, Jan 26 2016

Sum_{n>=0} a(n)/A013709(n) = 1 - 3/Pi (see A089491).

Corrected and extended by Vincenzo Librandi, Jan 26 2016

A284059 The absolute values of A275966.

Original entry on oeis.org

1, 1, 2, 0, 0, 2, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 4, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 4, 0, 2, 0, 0, 2, 4, 0, 0, 4, 2, 0, 4, 0, 2, 0, 0, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0

Offset: 1

Gevorg Hmayakyan, Mar 19 2017

This is multiplicative function with a(p^n) = |Re(I^(p^n+1) - I^(p^(n-1)+1))|.

			a(9) = |Re(I*(mobius(1)*I^9 + mobius(3)*I^3 + mobius(9)*I))| = |Re((I^10 - I^4))| = |-2| = 2.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A089491, A275966.

a(n):=abs(Re(I*add(numtheory:-mobius(d)*I^(n/d), d = numtheory:-divisors(n)))).

Table[Abs@ Re[I* Sum[MoebiusMu[d] * I^(n/d), {d, Divisors[n]}]], {n, 87}] (* Indranil Ghosh, Mar 19 2017 *)
f[p_, e_] := If[Mod[p, 4] == 1, 0, 2]; f[2, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 27 2024 *)

a(n)=my(f=factor(n)); prod(i=1,#f~, if(f[i,1]==2, if(f[i,2]==1,1,0), if(f[i,1]%4==3, 2, 0))) \\ Charles R Greathouse IV, Mar 22 2017

a(n) = |Re(I*Sum_{d|n}(mobius(d)*I^(n/d)))|.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/Pi = 0.954929... (A089491). - Amiram Eldar, Jan 27 2024

A371604 Decimal expansion of 5 * sqrt(3 - phi) / (2 * Pi).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Apr 01 2024

			0.93548928378863903321291906615298281...

Cf. A019845, A060294, A086089, A089491, A094874, A112628, A182007, A258403.

RealDigits[5 Sqrt[3 - GoldenRatio]/(2 Pi), 10, 99][[1]]

Equals Product_{k>=1} (1 - 1/(5*k)^2).

Equals A258403/Pi. - Hugo Pfoertner, Apr 01 2024

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

A132709 Decimal expansion of 19/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 31 2007

			6.04788783749202275921758300815554575730946653813734505241135907423807831010....

Index entries for transcendental numbers

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

Digits:=100; evalf(19/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[19/Pi,6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

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

More terms from Vladimir Joseph Stephan Orlovsky, Dec 02 2009

cp's OEIS Frontend

A132721 Decimal expansion of 31/Pi.

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A132717 Decimal expansion of 27/Pi.

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A268085 a(n) = Catalan(n)^2*n.

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Magma

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A284059 The absolute values of A275966.

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A371604 Decimal expansion of 5 * sqrt(3 - phi) / (2 * Pi).

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A132703 Decimal expansion of 13/Pi.

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A132704 Decimal expansion of 14/Pi.

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