A089491 -id:A089491 - cp's OEIS frontend

A132696 Decimal expansion of 6/Pi.

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

Offset: 1

Omar E. Pol, Aug 26 2007, Nov 02 2007

6/Pi = Volume of the cuboid (If L1>L2>L3) / Volume of the inscribed ellipsoid.
6/Pi = Volume of the cuboid (If L1>(L2=L3)) / Volume of the inscribed spheroid.
6/Pi = Volume of the regular hexahedron (or cube) / Volume of the inscribed Sphere.
6/Pi = 1 / Arc of 30 degrees.
6/Pi = Volume of the cuboid (If L1<(L2=L3)) / Volume of the inscribed spheroid.
6/Pi = Surface area of the regular hexahedron (or cube) / surface area of the inscribed sphere.

			1.90985931710274402922660516047... .

Index entries for transcendental numbers

Cf. A049541, A060294, A089491, A088538, A086201, A059956.

RealDigits[N[6/Pi, 200]] (* Erich Friedman, Mar 22 2008 *)

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

Equals Product_{k>=1} (2k+1)^3 / ( (2k)^2*(2k+3) ). - Federico Provvedi, Nov 09 2024

More terms from Erich Friedman, Mar 22 2008

A132702 Decimal expansion of 12/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

From Bernard Schott, Apr 17 2022: (Start)
For any triangle ABC, (see Crux Mathematicorum):
(b+c)/A + (c+a)/B + (a+b)/C >= (12/Pi) * s,
b*c/(A*(s-a)) + c*a/(B*(s-b)) + a*b/(C*(s-c)) >= (12/Pi) * s,
where (A,B,C) are the angles (measured in radians), (a,b,c) the side lengths of this triangle and s the semiperimeter.
Equality stands iff triangle ABC is equilateral. (End)

			3.819718634...

S. Arslanagić and D. M. Milošević, Problem 1827, Crux Mathematicorum, Vol. 22, No. 1 (1996), p. 36.
Index entries for transcendental numbers.

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

Digits:=100; evalf(12/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[12/Pi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2009 *)

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

Equals 2*A132696 = 4*A089491 = 6*A060294. -R. J. Mathar, Jul 29 2024

More terms from Vladimir Joseph Stephan Orlovsky, Jun 19 2009

A132699 Decimal expansion of 9/Pi.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Aug 26 2007

9/Pi = 2.864788975654...

Index entries for transcendental numbers

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

Digits:=100; evalf(9/Pi); # Wesley Ivan Hurt, Mar 26 2014

RealDigits[N[9/Pi,6! ]][[1]] (* Vladimir Joseph Stephan Orlovsky, Jun 19 2009 *)

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

More terms from Vladimir Joseph Stephan Orlovsky, Jun 19 2009

A132701 Decimal expansion of 11/Pi.

Original entry on oeis.org

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

A091379 a(n) = Product_{ p | n } (1 + Legendre(-1,p) ).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, Mar 02 2004

Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2) (but without the restriction that a(4k) = 0 and with a different definition of Legendre(-1,2)).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000086, A000091, A034444, A089491, A091392, A091393, A091394, A091395, A091396, A091397, A091398, A091399, A091400.

with(numtheory); A091379 := proc(n) local i,t1,t2; t1 := ifactors(n)[2]; t2 := mul((1+legendre(-1,t1[i][1])),i=1..nops(t1)); end;

a[n_] := Module[{t1, t2}, t1 = FactorInteger[n]; t2 = Product[(1 + KroneckerSymbol[-1, t1[[i, 1]]]), {i, 1, Length[t1]}]]; a[1] = 1;
Array[a, 105] (* Jean-François Alcover, Feb 08 2022, from Maple code *)

vecproduct(v) = { my(m=1); for(i=1,#v,m *= v[i]); m; };
A091379(n) = vecproduct(apply(p -> (1 + kronecker(-1,p)), factorint(n)[, 1])); \\ Antti Karttunen, Nov 18 2017

Here we use the definition that Legendre(-1, 2) = 1, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4.
From Amiram Eldar, Oct 11 2022: (Start)
Multiplicative with a(p^e) = 0 if p == 3 (mod 4) and 2 otherwise.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/Pi = 0.954929... (A089491). (End)

A016910 a(n) = (6*n)^2.

Original entry on oeis.org

0, 36, 144, 324, 576, 900, 1296, 1764, 2304, 2916, 3600, 4356, 5184, 6084, 7056, 8100, 9216, 10404, 11664, 12996, 14400, 15876, 17424, 19044, 20736, 22500, 24336, 26244, 28224, 30276, 32400, 34596, 36864, 39204, 41616, 44100, 46656, 49284, 51984, 54756, 57600, 60516, 63504, 66564, 69696, 72900

Offset: 0

N. J. A. Sloane

Areas A of two classes of triangles with integer sides (a,b,c) where a = 9k, b=10k and c = 17k, or a = 3k, b = 25k and c = 26k for k=0,1,2,... These areas are given by Heron's formula A = sqrt(s(s-a)(s-b)(s-c)) = (6k)^2, with the semiperimeter s = (a+b+c)/2. This sequence is a subsequence of A188158. - Michel Lagneau, Oct 11 2013

Sequence found by reading the line from 0, in the direction 0, 36, ..., in the square spiral whose vertices are the generalized 20-gonal numbers A218864. - Omar E. Pol, May 13 2018.

Vincenzo Librandi, Table of n, a(n) for n = 0..2000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000217, A089491, A188158, A243941.

Cf. similar sequences of the type k*n^2: A000290 (k=1), A001105 (k=2), A033428 (k=3), A016742 (k=4), A033429 (k=5), A033581 (k=6), A033582 (k=7), A139098 (k=8), A016766 (k=9), A033583 (k=10), A033584 (k=11), A135453 (k=12), A152742 (k=13), A144555 (k=14), A064761 (k=15), A016802 (k=16), A244630 (k=17), A195321 (k=18), A244631 (k=19), A195322 (k=20), A064762 (k=21), A195323 (k=22), A244632 (k=23), A195824 (k=24), A016850 (k=25), A244633 (k=26), A244634 (k=27), A064763 (k=28), A244635 (k=29), A244636 (k=30).

[(6*n)^2: n in [0..40]]; // Vincenzo Librandi, May 03 2011

(6*Range[0,40])^2 (* or *) LinearRecurrence[{3,-3,1},{0,36,144},40] (* Harvey P. Dale, Dec 25 2017 *)
Table[36 n^2, {n, 0, 45}] (* Omar E. Pol, Jun 07 2018 *)

a(n)=36*n^2 \\ Charles R Greathouse IV, Jun 10 2016

From Ilya Gutkovskiy, Jun 09 2016: (Start)
O.g.f.: 36*x*(1 + x)/(1 - x)^3.
E.g.f.: 36*x*(1 + x)*exp(x).
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
Sum_{n>=1} 1/a(n) = Pi^2/216 = A086726. (End)
Product_{n>=1} a(n)/A136017(n) = Pi/3. - Fred Daniel Kline, Jun 09 2016
a(n) = t(9*n) - 9*t(n), where t(i) = i*(i+k)/2 for any k. Special case (k=1): a(n) = A000217(9*n) - 9*A000217(n). - Bruno Berselli, Aug 31 2017
a(n) = 36*A000290(n) = 18*A001105(n) = 12*A033428 = 9*A016742(n) = 6*A033581(n) = 4*A016766(n) = 3*A135453(n) = 2*A195321(n). - Omar E. Pol, Jun 07 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi^2/432. - Amiram Eldar, Jun 27 2020
From Amiram Eldar, Jan 25 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = sinh(Pi/6)/(Pi/6).
Product_{n>=1} (1 - 1/a(n)) = sin(Pi/6)/(Pi/6) = 3/Pi (A089491). (End)

A112628 Decimal expansion of 2*sqrt(2)/Pi.

Original entry on oeis.org

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

Offset: 0

Rick L. Shepherd, Jan 11 2006

Example of extension to Buffon's Needle Problem: The probability that the boundary of a square will intersect one of the parallel lines if the square's diagonal length l (almost) equals the distance d between each pair of lines. This follows directly from the Weisstein/MathWorld Buffon's Needle Problem link's statement P=p/(Pi*d), where P is the probability of intersection with any convex polygon's boundary if the generalized diameter of that polygon is less than d and p is the perimeter of the polygon. (Take d=l, then p=2*sqrt(2)*d.).

The area of a regular octagon circumscribed in a unit-area circle. - Amiram Eldar, Nov 05 2020

			0.9003163161571060695551991910067405826645741499552206255714374712314587307...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Buffon's needle problem.
Eric Weisstein's World of Mathematics, Generalized Diameter.
Index entries for transcendental numbers

Cf. A060294 (2/Pi), A089491 (3/Pi), A224268.

R:= RealField(100); 2*Sqrt(2)/Pi(R); // G. C. Greubel, Aug 17 2018

RealDigits[2 Sqrt[2]/Pi, 10, 110][[1]] (* Bruno Berselli, Apr 02 2013 *)
(* From the second comment: *) RealDigits[N[Product[1 - 1/(4 n)^2, {n, 1, Infinity}], 110]][[1]] (* Bruno Berselli, Apr 02 2013 *)

PARI
```
2*sqrt(2)/Pi
```

Equals Product_{n>=1} (1-1/(4*n)^2). - Bruno Berselli, Apr 02 2013
Equals sinc(Pi/4). - Peter Luschny, Oct 04 2019
Equals Product_{k>=3} cos(Pi/2^k). - Amiram Eldar, Aug 24 2020

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

A132697 Decimal expansion of 7/Pi.

Original entry on oeis.org

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

cp's OEIS Frontend

A132696 Decimal expansion of 6/Pi.

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A132702 Decimal expansion of 12/Pi.

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A132699 Decimal expansion of 9/Pi.

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A132701 Decimal expansion of 11/Pi.

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A091379 a(n) = Product_{ p | n } (1 + Legendre(-1,p) ).

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A016910 a(n) = (6*n)^2.

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A112628 Decimal expansion of 2*sqrt(2)/Pi.

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