A212002 -id:A212002 - cp's OEIS frontend

Showing 1-10 of 12 results. Next

A196737 Decimal expansion of (4*Pi^2)/sqrt(35) = A212002/A010490.

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

Offset: 1

Raphie Frank, Dec 21 2012

This sequence is exactly equal to (Pi*h)/Spin(5/2), where h = Planck's Constant = A003676 and Spin(n/2) = h/(4*Pi) * sqrt(n(n+2)) = A003676/(10*A019694) * sqrt(A005563(n)).

			6.673070521654371272396016391388419924371668300691857264579256516590541...

Wikipedia, Spin (physics)

Cf. A212002, A010490.

RealDigits[(4Pi^2)/Sqrt[35],10,120][[1]] (* Harvey P. Dale, Jan 26 2021 *)

A212003 Decimal expansion of (2*Pi)^3.

Original entry on oeis.org

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

Offset: 3

Omar E. Pol, Aug 11 2012

			248.05021344239856140381052053681116161780230852708...

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998, p. 5.
Index entries for transcendental numbers

Cf. A000796, A019692, A091925, A212002, A212004.

RealDigits[(2*Pi)^3,10,120][[1]] (* Harvey P. Dale, Oct 09 2013 *)

8*Pi^3 \\ Charles R Greathouse IV, Oct 02 2022

Equals Product_{k=1..14, gcd(k,14)==1} Gamma(k/14). - Amiram Eldar, Jun 12 2021

A228746 Expansion of 8 * phi(q)^4 - 7 * phi(-q)^4 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 120, 24, 480, 24, 720, 96, 960, 24, 1560, 144, 1440, 96, 1680, 192, 2880, 24, 2160, 312, 2400, 144, 3840, 288, 2880, 96, 3720, 336, 4800, 192, 3600, 576, 3840, 24, 5760, 432, 5760, 312, 4560, 480, 6720, 144, 5040, 768, 5280, 288, 9360, 576, 5760, 96, 6840

Offset: 0

Michael Somos, Sep 02 2013

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Convolution with A005875 is A004008.

			G.f. = 1 + 120*q + 24*q^2 + 480*q^3 + 24*q^4 + 720*q^5 + 96*q^6 + 960*q^7 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A004008, A004011, A005875, A008438, A212002, A228745.

Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma0(4), 2), 50); A[1] + 120*A[2]; /* Michael Somos, Aug 21 2014 */

a[ n_] := SeriesCoefficient[ (8 EllipticTheta[ 3, 0, q]^4 - 7 EllipticTheta[ 4, 0, q]^4), {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( 8 * A^4 - 7 * subst(A, x, -x)^4, n))};

a(n) = 120 * b(n) with b() multiplicative where b(2^e) = 1/5 if e>1, b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)), if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 32 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A228745.
G.f.: 8 * (Sum_{k in Z} x^k^2)^4 - 7 * (Sum_{k in Z} (-x)^k^2)^4 .
a(2*n) = A004011(n). a(2*n + 1) = 120 * A008438(n).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 4*Pi^2 = 39.478417... (A212002). - Amiram Eldar, Dec 29 2023

A212004 Decimal expansion of (2*Pi)^4.

Original entry on oeis.org

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

Offset: 4

Omar E. Pol, Aug 11 2012

			1558.545456544038995783...

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998, p. 11.
Index entries for transcendental numbers

Cf. A000796, A019692, A092425, A212002, A212003, A212005.

RealDigits[(2*Pi)^4, 10, 100][[1]] (* Amiram Eldar, Jun 12 2021 *)

16*Pi^4 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{k=1..15, gcd(k,15)==1} Gamma(k/15). - Amiram Eldar, Jun 12 2021

A212005 Decimal expansion of (2*Pi)^5.

Original entry on oeis.org

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

Offset: 4

Omar E. Pol, Aug 11 2012

			9792.629913129006504407721921389939407369622612098399...

Index entries for transcendental numbers

Cf. A000796, A019692, A092731, A212002, A212003, A212004, A212006.

RealDigits[(2*Pi)^5, 10, 100][[1]] (* Amiram Eldar, Jun 12 2021 *)

32*Pi^5 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{k=1..22, gcd(k,22)==1} Gamma(k/22). - Amiram Eldar, Jun 12 2021

A212006 Decimal expansion of (2*Pi)^6.

Original entry on oeis.org

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

Offset: 5

Omar E. Pol, Aug 11 2012

			61528.908388819483969934...

Juan M. Maldacena, The Large N Limit of Superconformal Field Theories and Supergravity, arXiv:hep-th/9711200v3, 1997-1998, p. 13, 15.
Index entries for transcendental numbers

Cf. A000796, A019692, A092732, A212002-A212005.

RealDigits[(2*Pi)^6, 10, 100][[1]] (* Amiram Eldar, Jun 12 2021 *)

64*Pi^6 \\ Charles R Greathouse IV, Oct 01 2022

Equals Product_{k=1..21, gcd(k,21)==1} Gamma(k/21). - Amiram Eldar, Jun 12 2021

A233700 Decimal expansion of 1/sin(arctan(1/t)) or t/sin(arctan(t)) where t = 2*Pi: hypotenuse for a right triangle of equal area to a disk.

Original entry on oeis.org

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

Offset: 1

John W. Nicholson, Dec 16 2013

"The great mathematician Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle." (Quote from Wikipedia link)

			6.362265131567328393691245440586804410699714985138989686582041617045998587331...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Wikipedia, Area of a disk: Triangle method

Cf. A233527, A019692, A233528.

using Nemo
RR = RealField(310)
t = const_pi(RR) + const_pi(RR)
t/sin(atan(t)) |> println # Peter Luschny, Mar 13 2018

C := ComplexField(); Sqrt(1 + 4*Pi(C)^2) // G. C. Greubel, Jan 08 2018

Magma

R:=RealField(110); SetDefaultRealField(R); n:=Sqrt(1+4*Pi(R)^2); Reverse(Intseq(Floor(10^108*n))); // Bruno Berselli, Mar 13 2018

Mathematica

RealDigits[(2*Pi)/Sin[ArcTan[2*Pi]],10,120][[1]] (* Harvey P. Dale, Jul 12 2014 *) RealDigits[ Sqrt[1 + 4*Pi^2], 10, 111][[1]] (* Robert G. Wilson v, Mar 12 2015 *)

PARI

sqrt(1+(2*Pi)^2)

Formula

Equals sqrt(1+(2*Pi)^2) = sqrt(1 + (A019692)^2) = sqrt(1 + A212002) = 1/sin(A233527) = A019692/sin(A233528) = 1/cos(A233528) = A019692/cos(A233527).

A275322 Decimal expansion of AGM(1, sqrt(2))^2/Pi.

Original entry on oeis.org

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

Offset: 0

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Author

Dimitris Valianatos, Jul 23 2016

Keywords

nonn
cons

Comments

Conjecture: Equals Product_{n odd} (n/(n+2) if n == 1 (mod 4), (n+2)/n otherwise) = (1/3) * (5/3) * (5/7) * (9/7) * (9/11) * (13/11) * (13/15) * (17/15) * (17/19) * (21/19) * (21/23) * (25/23) * (25/27) * ...

Examples

0.45694658104446362537496662254768...

Links

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

Markus Faulhuber, Anupam Gumber, and Irina Shafkulovska, The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators, arXiv:2209.04202 [math.CA], 2022, p. 21.

Crossrefs

Cf. A053004 (AGM(1, sqrt(2))).

Cf. A000796, A002161, A068466, A212002.

Programs

Magma

SetDefaultRealField(RealField(100)); R:= RealField(); 8*Pi(R)^2/Gamma(1/4)^4; // G. C. Greubel, Oct 07 2018

Maple

evalf(GaussAGM(1,sqrt(2))^2/Pi,100); # Muniru A Asiru, Oct 08 2018

Mathematica

First@ RealDigits@ N[ArithmeticGeometricMean[1, Sqrt[2]]^2/Pi, 120] (* Michael De Vlieger, Jul 26 2016 *)

PARI

agm(1, sqrt(2)) ^ 2 / Pi

PARI

8*Pi^2/gamma(1/4)^4 \\ Altug Alkan, Oct 08 2018

Formula

Equals 8*Pi^2/Gamma(1/4)^4 = 4*Gamma(3/4)^2/Gamma(1/4)^2. - Vaclav Kotesovec, Sep 22 2016

A212007 Decimal expansion of (2*Pi)^7.

Original entry on oeis.org

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

Offset: 6

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Author

Omar E. Pol, Aug 13 2012

Keywords

nonn
cons

Examples

386597.533155429384641818431113221354485431356837811...

Links

J. M. Maldacena, The gauge/gravity duality, arXiv:1106.6073, p. 14.

Index entries for transcendental numbers

Crossrefs

Cf. A000796, A019692, A092735, A212002-A212006.

Programs

Mathematica

RealDigits[(2*Pi)^7,10,120][[1]] (* Harvey P. Dale, Jun 26 2019 *)

PARI

(2*Pi)^7 \\ Charles R Greathouse IV, Oct 21 2013

A377594 Decimal expansion of 1/4 - 7*zeta(3)/(4*Pi^2).

Original entry on oeis.org

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

Offset: 0

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Author

Stefano Spezia, Nov 02 2024

Keywords

nonn
cons

Examples

0.036860800591247104538239286701916563470996615185...

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6.3, p. 44.

Links

Michael I. Shamos, A catalog of the real numbers, (2007). See p. 69.

Crossrefs

Cf. A000302, A002117, A005408, A212002.

Programs

Mathematica

RealDigits[1/4-7Zeta[3]/(4Pi^2),10,100][[1]]

Formula

Equals Sum_{k>=1} zeta(2*k)/((2*k + 1)*(2*k + 2)*2^(2*k)) [Euler] (see Finch and Shamos).

Showing 1-10 of 12 results. Next

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A196737 Decimal expansion of (4*Pi^2)/sqrt(35) = A212002/A010490.

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Mathematica

A212003 Decimal expansion of (2*Pi)^3.

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Mathematica

PARI

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A228746 Expansion of 8 * phi(q)^4 - 7 * phi(-q)^4 in powers of q where phi() is a Ramanujan theta function.

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Magma

Mathematica

PARI

Formula

A212004 Decimal expansion of (2*Pi)^4.

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Mathematica

PARI

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A212005 Decimal expansion of (2*Pi)^5.

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Mathematica

PARI

Formula

A212006 Decimal expansion of (2*Pi)^6.

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Mathematica

PARI

Formula

A233700 Decimal expansion of 1/sin(arctan(1/t)) or t/sin(arctan(t)) where t = 2*Pi: hypotenuse for a right triangle of equal area to a disk.

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Julia

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Magma

A377594 Decimal expansion of 1/4 - 7zeta(3)/(4Pi^2).