A223067 -id:A223067 - cp's OEIS frontend

A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n).

1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 11, 17, 1, 23, 1, 11, 1, 1, 1, 17, 11, 1, 1, 1, 23, 11, 43, 17, 1, 1, 121, 1, 1, 1, 1, 4301, 1, 1, 1, 73, 11, 1, 1, 17, 1, 11, 23, 43, 1, 1, 11, 17, 1, 1, 1, 11, 101, 23, 89, 17, 11, 1, 1, 83, 1, 11, 1

Offset: 0

Johannes W. Meijer and Joseph Abate, Jan 03 2017

This sequence is related in a peculiar way to A223067 and A223068, sequences related to the complete elliptic integral of the first kind K(k), and to A280442 and A046161, sequences related to the unsigned Euler numbers A000364.
In this sequence certain prime numbers appear on a regular basis, either by itself or as a factor of a composite number, i.e., a(n)=11 if n=7+5*k, a(n)=17 if n=13+8*k, a(n)=23 if n=15+11*k, a(n)=43 if n=28+21*k, a(n)=73 if n=41+36*k, a(n)=101 if n=58+50*k, a(n)=89 if n=60+44*k, a(n)=83 if n=65+41*k, in all cases k >= 0. We observe that the period T of each prime is apparently T = (prime-1)/2.
Conjecture: The sequence A280443 will not have a(n)=1 after some point.

Cf. A000364 (Euler numbers), A046161, A067624, A223067, A223068, A280442.

nmax:=68: A067624 := n -> 2^(2*n)*(2*n)!: f := series((exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1))), x=0, nmax+1): for n from 0 to nmax do b(n) := coeff(f, x, n); a(n) := numer(b(n))/numer(b(n)/A067624(n)) od: seq(a(n), n=0..nmax);

def A280443_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    g = lambda x: exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2*n - sum(n.digits(2))
    return [(2^d(n)*R[n]/(numerator(R[n]/factorial(2*n)))) for n in (0..prec)]
print(A280443_list(68)) # Peter Luschny, Jan 05 2017

a(n) = A280442(n)/A223067(n).
a(n) = A067624(n)*A046161(n)/A223068(n).
a(n) = A280442(n)/numer((A280442(n)/A046161(n))/A067624(n)).

A019692 Decimal expansion of 2*Pi.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Pi/5 or 2*Pi/10 is the expected surface area containing completely a Brownian curve (trajectory) on a plane. - Lekraj Beedassy, Jul 28 2005
Bob Palais considers this a more fundamental constant than Pi, see the Palais reference and link. - Jonathan Vos Post, Sep 10 2010
The Persian mathematician Jamshid al-Kashi seems to have been the first to use the circumference divided by the radius as the circle constant. In Treatise on the Circumference published 1424 he calculated the circumference of a unit circle to 9 sexagesimal places. - Peter Harremoës, John W. Nicholson, Aug 02 2012
"Proponents of a new mathematical constant tau (τ), equal to two times π, have argued that a constant based on the ratio of a circle's circumference to its radius rather than to its diameter would be more natural and would simplify many formulas" (from Wikipedia). - Jonathan Sondow, Aug 15 2012
The constant 2*Pi appears in the formula for the period T of a simple gravity pendulum. For small angles this period is given by Christiaan Huygens’s law, i.e., T = 2*Pi*sqrt(L/g), see for more information A223067. - Johannes W. Meijer, Mar 14 2013
There are seven consecutive nines at positions 762 to 768. - Roland Kneer, Jul 05 2013
Volume of a cylinder in which a sphere of radius 1 can be inscribed. - Omar E. Pol, Sep 25 2013
2*Pi is also the surface area of a sphere whose diameter equals the square root of 2. More generally, x*Pi is also the surface area of a sphere whose diameter equals the square root of x. - Omar E. Pol, Dec 18 2013
From Bernard Schott, Jan 31 2020: (Start)
Also, (2*Pi)*a^2 is the area of the deltoid (an hypocycloid with three cusps) whose Cartesian parametrization is:
   x = a * (2*cos(t) + cos(2*t)),
   y = a * (2*sin(t) - sin(2*t)).
The length of this deltoid is 16*a. See the curve at the Mathcurve link. (End)
Pi/5 = 0.1 * 2*Pi is the mean area of the plane triangles formed by 3 points independently and uniformly chosen at random on the surface of a unit-radius sphere. - Amiram Eldar, Aug 06 2020

			6.283185307179586476925286766559005768394338798750211641949889184615632...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.4, p. 17.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 69.

Harry J. Smith, Table of n, a(n) for n = 1..20000
Robert Ferréol, Deltoid, Mathcurve.
Christophe Garban and José A. Trujillo Ferreras, The expected area of the filled planar Brownian loop is pi/5, Communications in mathematical physics, Vol. 264, No. 3 (2006), pp. 797-810, preprint, arXiv:math/0504496 [math.PR], 2005.
Peter Harremoës, Al-Kashi’s constant
Michael Hartl, The Tau Manifesto.
Melissa Larson, Verifying and discovering BBP-type formulas, 2008.
Bob Palais, Web page about "Pi is wrong!".
Bob Palais, Pi is wrong!, The Mathematical Intelligencer Volume 23, Number 3, 2001, pp. 7-8.
Grant Sanderson, How pi was almost 6.283185..., 3Blue1Brown video (2018).
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 318.
Wikipedia, Tau proposals.
Wikipedia, Bailey-Borwein-Plouffe formula.
Index entries for transcendental numbers.

Cf. A058291 (continued fraction).
Cf. A000796, A019693, A019699, A091925, A244979.
Cf. A093828 (astroid), A180434 (loop of strophoid), A197723 (cardioid).

using Nemo
RR = RealField(334)
tau = const_pi(RR) + const_pi(RR)
tau |> println # Peter Luschny, Mar 14 2018

R:= RealField(100); 2*Pi(R); // G. C. Greubel, Mar 08 2018

RealDigits[N[2 Pi, 6! ]] (* Vladimir Joseph Stephan Orlovsky, Dec 02 2009 *)

default(realprecision, 20080); x=2*Pi; for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b019692.txt", n, " ", d)); \\ Harry J. Smith, May 31 2009

# Use some guard digits when computing.
# BBP formula P(1, 16, 8, (0, 8, 4, 4, 0, 0, -1, 0)).
from decimal import Decimal as dec, getcontext
def BBPtau(n: int) -> dec:
    getcontext().prec = n
    s = dec(0); f = dec(1); g = dec(16)
    for k in range(n):
        ek = dec(8 * k)
        s += f * ( dec(8) / (ek + 2) + dec(4) / (ek + 3)
                 + dec(4) / (ek + 4) - dec(1) / (ek + 7))
        f /= g
    return s
print(BBPtau(200))  # Peter Luschny, Nov 03 2023

e^(Zeta'(0)/Zeta(0)) = 2*Pi. - Peter Luschny, Jun 17 2018
From Peter Bala, Oct 30 2019: (Start)
2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/6) + 1/(n + 5/6) ).
2*Pi = Sum_{n >= 0} (-1)^n*( 1/(n + 1/10) - 1/(n + 3/10) - 1/(n + 7/10) + 1/(n + 9/10) ). Cf. A091925 and A244979. (End)
From Amiram Eldar, Aug 06 2020: (Start)
Equals Gamma(1/6)*Gamma(5/6).
Equals Integral_{x=0..oo} log(1 + 1/x^6) dx.
Equals Integral_{x=0..oo} log(1 + 4/x^2) dx.
Equals Integral_{x=-oo..oo} exp(x/6)/(exp(x) + 1) dx.
Equals Sum_{k>=0} 1/((k + 1/4)*(k + 3/4)). (End)
Equals 4*Integral_{x=0..1} 1/sqrt(1 - x^2) dx (see Finch). - Stefano Spezia, Oct 19 2024

A067624 a(n) = 2^(2n)(2*n)!.

Original entry on oeis.org

1, 8, 384, 46080, 10321920, 3715891200, 1961990553600, 1428329123020800, 1371195958099968000, 1678343852714360832000, 2551082656125828464640000, 4714400748520531002654720000, 10409396852733332453861621760000

Offset: 0

Benoit Cloitre, Feb 02 2002

nonn

For n >= 1, a(n) equals the absolute value of the determinant of the 4n X 4n matrix with i's along the superdiagonal (where i is the imaginary unit), and 2, 3, 4, ... 4*n along the subdiagonal, and 0's everywhere else. (See Mathematica code below.) - John M. Campbell, Jun 04 2011

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A000165.

Appears in A162445, A061549 and A120738. - Johannes W. Meijer, Jul 06 2009

[2^(2*n)*Factorial(2*n): n in [0..15]]; // Vincenzo Librandi, Feb 18 2018

for n from 0 to 30 by 2 do printf(`%d,`,2^(n)*(n)!) od: # James Sellers, Feb 11 2002
A067624 := n -> 2^(2*n)*(2*n)!: seq(A067624(n), n=0..12); # Johannes W. Meijer, Jan 05 2017

Table[Abs[Det[Array[KroneckerDelta[#1 + 1, #2]*I &, {4*n, 4*n}] + Array[KroneckerDelta[#1 - 1, #2]*#1 &, {4*n, 4*n}]]], {n, 1, 20}] (* John M. Campbell, Jun 04 2011 *)
Table[2^(2 n) (2 n)!, {n, 0, 30}] (* Vincenzo Librandi, Feb 18 2018 *)

a(n) = A000165(2*n) where A000165(k) are the double factorial numbers 2^k*k!=(2k)!!. - Corrected by Johannes W. Meijer, Jul 05 2009
a(n) = (4*n)!! = 2^(2*n)*(2*n)!. - Johannes W. Meijer, Jul 06 2009
sqrt((1+cos(x))/2) = Sum_{n>=0} (-1)^n * x^(2*n) / a(n).
a(n) = (A280442(n)/A046161(n))/(A223067(n)/A223068(n)). - Johannes W. Meijer, Jan 05 2017
From Amiram Eldar, Jul 12 2020: (Start)
Sum_{n>=0} 1/a(n) = cosh(1/2).
Sum_{n>=0} (-1)^n/a(n) = cos(1/2). (End)

More terms from James Sellers, Feb 11 2002

A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^neuler(2n)x^n/(2n)).

Original entry on oeis.org

1, 1, 11, 173, 22931, 1319183, 233526463, 29412432709, 39959591850371, 8797116290975003, 4872532317019728133, 1657631603843299234219, 2718086236621937756966743, 1321397724505770800453750299, 1503342018433974345747514544039

Offset: 0

Johannes W. Meijer and Joseph Abate, Jan 03 2017

This sequence is related in a peculiar way to A223067, a sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes. See A280443 for more information.

Sergey Khrushchev, Orthogonal Polynomials and Continued Fractions, From Euler's point of view, Corollary 4.26, p. 192, 2008.

Cf. A046161 (denominators).

Cf. A000364 (Euler numbers), A223067, A255881, A280443.

nmax:=14: f := series(exp(add((-1)^n*euler(2*n) * x^n/(2*n), n=1..nmax+1)), x=0, nmax+1): for n from 0 to nmax do a(n) := numer(coeff(f, x, n)) od: seq(a(n), n=0..nmax);

def A280442_list(prec):
    P. = PowerSeriesRing(QQ, default_prec=2*prec)
    def g(x): return exp(sum((-1)^k*euler_number(2*k)*x^k/(2*k) for k in (1..prec+1)))
    R = P(g(x)).coefficients()
    d = lambda n: 2^(2*n - sum(n.digits(2)))
    return [d(n)*R[n] for n in (0..prec)]
print(A280442_list(14)) # Peter Luschny, Jan 05 2017

a(n) = numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n * euler(2*n)*x^n/(2*n)).

Let S = Sum_{n>=0} (-1)^n*euler(2*n)*x^n/(2*n) and w(n) = A005187(n) then a(n) = 2^w(n) * [x^n] exp(S). - Peter Luschny, Jan 05 2017

A223068 A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes.

Original entry on oeis.org

1, 16, 3072, 737280, 1321205760, 951268147200, 2009078326886400, 265928913086054400, 44931349155019751424000, 109991942731488351485952000, 668751011807449177034588160000, 2471703739640332158319837839360000

Offset: 0

Johannes W. Meijer, Mar 14 2013

The period T of a simple gravity pendulum for arbitrary amplitudes is given by a complicated formula, see A223067. The Taylor series expansion of T as a function of the angular displacement phi leads for the denominators of the even powers of phi to the sequence given above and for the numerators to A223067.

			T = 2*Pi*sqrt(L/g) * (1 + (1/16)*phi^2 + (11/3072)*phi^4 + (173/737280)*phi^6 + ... ).

Cf. A223067 (numerators), A019692 (2*Pi).

Cf. A280442, A280443.

nmax:=11: f := series(1/((Pi/4)*(1+cos(phi/2))/EllipticK((1-cos(phi/2))/(1+cos(phi/2)))), phi, 2*nmax+1): for n from 0 to nmax do a(n):= denom(coeff(f, phi, 2*n)) od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 05 2017

s = Series[EllipticK[Sin[t/2]^2 ], {t, 0, 50}]; CoefficientList[2*s, t^2] // Denominator (* Jean-François Alcover, Oct 07 2014 *)

cp's OEIS Frontend

A280443 a(n) = A280442(n)/A223067(n) = A067624(n)*A046161(n)/A223068(n).

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A019692 Decimal expansion of 2*Pi.

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A067624 a(n) = 2^(2*n)*(2*n)!.

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A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^n*euler(2*n)*x^n/(2*n)).

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Maple

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A223068 A sequence related to the period T of a simple gravity pendulum for arbitrary amplitudes.

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Maple

Mathematica

A067624 a(n) = 2^(2n)(2*n)!.

A280442 Numerators of coefficients in the Taylor series expansion of Sum_{n>=0} exp((-1)^neuler(2n)x^n/(2n)).