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.

A340322 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2) dz dy dx.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 04 2021

Keywords

Comments

Integral_{x=0..Pi/2, y=0..Pi/2} log(4*cos(x)^2 + 4*cos(y)^2) dy dx = G*Pi, where G is Catalan's constant A006752.

Examples

			6.485696465218497693708581372103315764152266325617976316831738842452555238784...

Links

Crossrefs

Cf. A071763, A097469, A340182, A340350, A340421, A340422.

Programs

  • Maple
    evalf(Integrate(log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2), x = 0..Pi/2, y = 0..Pi/2, z = 0..Pi/2));
  • PARI
    intnum(x = 0, Pi/2, intnum(y = 0, Pi/2, intnum(z = 0, Pi/2, log(4*cos(x)^2 + 4*cos(y)^2 + 4*cos(z)^2)))) \\ 20 valid digits

Formula

Equals limit_{n->infinity} Pi^3 * log(A340182(n)) / (8*n^3).
Equals Pi^3 * log(2)/8 + Integral_{x=0..Pi/2, y=0..Pi/2, z=0..Pi/2} log(3 + cos(2*x) + cos(2*y) + cos(2*z)) dz dy dx.

A340421 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2 + sin(y)^2) dy dx.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 07 2021

Keywords

Examples

			1.627008991085721315763766677017604437985734719035793082916212355323520769...

Crossrefs

Cf. A340396, A340322, A340350, A340422.

Programs

  • Mathematica
    RealDigits[N[-Pi^2*(Log[2] + Log[Sqrt[2] - 1]/2) + Pi*Integrate[Log[1 + Sqrt[1 + 1/(1 + Sin[x]^2)]], {x, 0, Pi/2}], 120], 10, 110][[1]]

Formula

Equals -Pi^2*(log(2) + log(sqrt(2)-1)/2) + Pi * Integral_{x=0..Pi/2} log(1 + sqrt(1 + 1/(1 + sin(x)^2))) dx.
Equals limit_{n->infinity} Pi^2 * (log(A340396(n))/n^2 - log(2)) / 4.

A340422 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dy dx.

Original entry on oeis.org

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

Views

Author

Vaclav Kotesovec, Jan 07 2021

Keywords

Examples

			2.551698806403924360993561678605629314325436926549295727591221339383517201757...

Crossrefs

Cf. A067518, A340322, A340350, A340421.

Programs

  • Mathematica
    RealDigits[N[Pi*Integrate[(Log[1 + Sqrt[1 + 2/(3 - 2*Cos[x]^2)]] + Log[(1 + 2*Sin[x]^2)/4]/2), {x, 0, Pi/2}], 120], 10, 110][[1]]

Formula

Equals Pi * Integral_{x=0..Pi/2} (log(1 + sqrt(1 + 2/(3 - 2*cos(x)^2))) + log((1 + 2*sin(x)^2)/4)/2) dx.
Equals limit_{n->infinity} Pi^2 * log(A067518(n))/(4*n^2) - Pi^2*log(2)/4 - G*Pi, where G is Catalan's constant A006752.

A340165 a(n) = 4^((n-2)*(n-1)) * Product_{1<=i

Original entry on oeis.org

1, 1, 19, 7056, 51251277, 7280323311888, 20225477546584790663, 1098876823994281426921193472, 1167619533875635661974056722756222809, 24263631353490502503207804571072304043237024000
Offset: 1

Views

Author

Seiichi Manyama, Dec 30 2020

Keywords

Crossrefs

Cf. A340139, A340167.

Programs

  • Mathematica
    Table[4^((n-2)*(n-1)) * Product[Product[1 + Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, j-1}], {j, 2, n-1}], {n, 1, 12}] // Round (* Vaclav Kotesovec, Dec 31 2020 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(4^((n-2)*(n-1))*prod(j=2, n-1, prod(i=1, j-1, 1+(sin(i*Pi/(2*n))*sin(j*Pi/(2*n)))^2)))}

Formula

a(n) = 4^((n-2)*(n-1)) * Product_{1<=i
a(n) ~ 2^(2*n^2 - 3*n + 35/8) * (1 - sqrt(2*sqrt(2)-2))^n * exp(2*A340350*n^2/Pi^2). - Vaclav Kotesovec, Jan 05 2021

A340167 a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 + sin(i*Pi/(2*n))^2 * sin(j*Pi/(2*n))^2).

Original entry on oeis.org

1, 20, 153425, 450075709440, 504979178328238519521, 216703205118496785026106198144000, 35568160616301682717925992221900586646216066081, 2232861039051291914755952483706805051795013026559178904468193280
Offset: 1

Views

Author

Seiichi Manyama, Dec 30 2020

Keywords

Crossrefs

Cf. A340165, A340166.

Programs

  • Mathematica
    Table[4^(2*(n-1)^2) * Product[Product[1 + Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, n-1}], {j, 1, n-1}], {n, 1, 10}] // Round (* Vaclav Kotesovec, Dec 31 2020 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(4^(2*(n-1)^2)*prod(i=1, n-1, prod(j=1, n-1, 1+(sin(i*Pi/(2*n))*sin(j*Pi/(2*n)))^2)))}

Formula

a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 + cos(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
a(n) = 4^(2*(n-1)^2) * Product_{1<=i,j<=n-1} (1 + sin(i*Pi/(2*n))^2 * cos(j*Pi/(2*n))^2).
a(n) ~ 2^(4*n^2 - 6*n + 17/4) * (sqrt(2) - 1)^(2*n) * exp(4*A340350*n^2/Pi^2). - Vaclav Kotesovec, Jan 05 2021
Showing 1-5 of 5 results.

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