cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A333198 Decimal expansion of a constant related to the asymptotics of A306734 and A333179.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Mar 11 2020

Keywords

Examples

			1.86429525435844065924747599856112246877295244507368421574403...

Crossrefs

Cf. A306734, A333179, A333180, A333181, A357471, A376621.

Programs

  • Mathematica
    RealDigits[E^Sqrt[4*Log[r]^2/3 + 4*PolyLog[2, 1-r] - Pi^2/3] /. r -> (2 - 5*(2/(-11 + 3*Sqrt[69]))^(1/3) + ((-11 + 3*Sqrt[69])/2)^(1/3))/3, 10, 120][[1]] (* Vaclav Kotesovec, Oct 07 2024 *)

Formula

Equals limit_{n->infinity} A306734(n)^(1/sqrt(n)).
Equals limit_{n->infinity} A333179(n)^(1/sqrt(n)).
Equals exp(sqrt(4*log(r)^2/3 + 4*polylog(2, 1-r) - Pi^2/3)), where r = 1 - A357471 = 0.4301597090019467340886... is the real root of the equation r^2 = (1-r)^3. - Vaclav Kotesovec, Oct 07 2024

Extensions

More digits from Vaclav Kotesovec, Oct 07 2024

A357470 Decimal expansion of the real root of x^3 - x^2 - 2*x - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Oct 25 2022

Keywords

Comments

This equals r0 + 1/3 where r0 is the real root of y^3 - (7/3)*y - 47/27.
The other roots of x^3 - x^2 - 2*x - 1 are (2 + w1*(4*(47 + 3*sqrt(93)))^(1/3) + w2*(4*(47 - 3*sqrt(93)))^(1/3))/6 = -0.5739495178... + 0.3689894074...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex roots of x^3 - 1.
Using hyperbolic functions these roots are (1 - sqrt(7)*(cosh((1/3)*arccosh((47/98)*sqrt(7))) - sqrt(3)*sinh((1/3)*arccosh((47/98)*sqrt(7)))*i))/3, and its complex conjugate.

Examples

			2.147899035704787354026214964930987364916766150370284279446911717889159675...

Crossrefs

Cf. A160389, A255524, A255249, A357471, A357472.

Programs

  • Mathematica
    RealDigits[x /. FindRoot[x^3 - x^2 - 2*x - 1, {x, 2}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)

Formula

r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + 28*(4*(47 + 3*sqrt(93)))^(-1/3))/6.
r = (2 + (4*(47 + 3*sqrt(93)))^(1/3) + (4*(47 - 3*sqrt(93)))^(1/3))/6.
r = (1 + 2*sqrt(7)*cosh((1/3)*arccosh((47/98)*sqrt(7))))/3.
r = (1/3) + (188^(1/3)/3)*Hyper2F1([-1/6, 1/3], [1/2], 837/(47^2)). - Gerry Martens, Nov 04 2022

A357472 Decimal expansion of the real root of x^3 + x^2 + 2*x - 1.

Original entry on oeis.org

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

Views

Author

Wolfdieter Lang, Oct 25 2022

Keywords

Comments

This equals r0 - 1/3 where r0 is the real root of y^3 + (5/3)*y - 43/27.
The other roots of x^3 - x^2 + 2*x - 1 are (-1 + w1*((43 + 9*sqrt(29))/2)^(1/3) + ((43 - 9*sqrt(29))/2)^(1/3))/3 = -0.6963233908... + 1.4359498641...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 is one of the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are -(1 + sqrt(5)*(sinh((1/3)*arcsinh( (43/50)*sqrt(5))) - sqrt(3)*cosh((1/3)*arcsinh((43/50)*sqrt(5)))*i))/3, and its complex conjugate.

Examples

			0.3926467817026408117648795948843412507037649685934825897311396498445171668...

Crossrefs

Cf. A160389, A255524, A255249, A357470, A357471.

Programs

  • Maple
    Digits:=100: u := 2/(43 + 9*sqrt(29)): (-5*u^(1/3) + u^(-1/3) - 1)/3:
evalf(%*10^78): ListTools:-Reverse(convert(floor(%), base, 10)); # Peter Luschny, Nov 01 2022
  • Mathematica
    RealDigits[x /. FindRoot[x^3 + x^2 + 2*x - 1, {x, 1}, WorkingPrecision -> 120]][[1]] (* Amiram Eldar, Oct 26 2022 *)

Formula

r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) - 20*(4*(43 + 9*sqrt(29)))^(-1/3))/6.
r = (-2 + (4*(43 + 9*sqrt(29)))^(1/3) + w1*(4*(43 - 9*sqrt(29)))^(1/3))/6, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i) is one of the complex conjugate roots of x^3 - 1.
r = (-1 + 2*sqrt(5)*sinh((1/3)*arcsinh((43/50)*sqrt(5))))/3.
r = (-5*u^(1/3) + u^(-1/3) - 1)/3 where u = 2/(43 + 9*sqrt(29)). - Peter Luschny, Nov 01 2022
r = (-1/3) + (43/45) * Hyper2F1([1/3, 2/3], [3/2], -43^2/(5*10^2)). - Gerry Martens, Nov 04 2022

A376708 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} 1/(1 - x^j)^3.

Original entry on oeis.org

1, 0, 1, 3, 6, 10, 16, 24, 37, 55, 84, 124, 186, 270, 394, 561, 798, 1114, 1553, 2133, 2924, 3966, 5364, 7196, 9629, 12795, 16956, 22344, 29355, 38377, 50026, 64920, 84006, 108275, 139155, 178207, 227601, 289734, 367882, 465726, 588147
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 02 2024

Keywords

Crossrefs

Cf. A003106, A064428, A376711.
Cf. A357471.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Sum[x^(k*(k+1))/Product[1-x^j, {j, 1, k}]^3, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ r^(1/3) * (log(r)^2 + 3*polylog(2, 1-r))^(3/4) * exp(2*sqrt((log(r)^2 + 3*polylog(2, 1-r))*n)) / (4 * Pi^(3/2) * sqrt(2+r) * n^(5/4)), where r = 1 - A357471 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3.

A376709 G.f.: Sum_{k>=0} x^(k^2) * Product_{j=1..k} 1/(1 - x^j)^3.

Original entry on oeis.org

1, 1, 3, 6, 11, 18, 30, 47, 75, 115, 177, 264, 394, 573, 831, 1184, 1679, 2349, 3273, 4511, 6192, 8428, 11422, 15372, 20606, 27453, 36435, 48103, 63270, 82833, 108068, 140399, 181806, 234541, 301636, 386604, 494080, 629459, 799770, 1013253, 1280463
Offset: 0

Views

Author

Vaclav Kotesovec, Oct 02 2024

Keywords

Crossrefs

Cf. A003114, A000041, A376712.
Cf. A357471.

Programs

  • Mathematica
    nmax = 40; CoefficientList[Series[Sum[x^(k^2)/Product[1-x^j, {j, 1, k}]^3, {k, 0, Sqrt[nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ (log(r)^2 + 3*polylog(2, 1-r))^(3/4) * exp(2*sqrt((log(r)^2 + 3*polylog(2, 1-r))*n)) / (4 * Pi^(3/2) * r^(2/3) * sqrt(2+r) * n^(5/4)), where r = 1 - A357471 = 0.4301597090019467340886... is the real root of the equation r^2 = (1-r)^3.
Showing 1-5 of 5 results.

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