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 1-3 of 3 results.

A340350 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

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

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Author

Vaclav Kotesovec, Jan 05 2021

Keywords

Examples

			0.49531661869212336430296504041161047588717884176797451824647459341123774...

Crossrefs

Cf. A097469, A340165, A340167, A340322, A340421, A340422.

Programs

  • Maple
    evalf(Pi * Integrate(log((1 + sqrt(1 + sin(x)^2))/2), x = 0..Pi/2), 120);
  • Mathematica
    RealDigits[N[Pi*Integrate[Log[(1 + Sqrt[1 + Sin[x]^2])/2], {x, 0, Pi/2}], 100]][[1]]
  • PARI
    Pi * intnum(x = 0, Pi/2, log((1 + sqrt(1 + sin(x)^2))/2))

Formula

Equals Pi * Integral_{x=0..Pi/2} log((1 + sqrt(1 + sin(x)^2))/2) dx.
Equals limit_{n->infinity} Pi^2 * (log(A340165(n)) / (2*n^2) - log(2)).
Equals limit_{n->infinity} Pi^2 * (log(A340167(n)) / (4*n^2) - log(2)).

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

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pi*j/n)^2 + sin(Pi*k/n)^2).

Original entry on oeis.org

0, 1, 96, 93789, 1244160000, 241885578271872, 700566272328037500000, 30323548995402141685610526683, 19627362048402730985830806120284160000, 189995156103157091521654945902925881881155376920, 27506190205802587152768139358989866456457087869970721213256
Offset: 0

Views

Author

Vaclav Kotesovec, Jan 06 2021

Keywords

Crossrefs

Cf. A004003, A007726, A065072, A067518, A127605, A340052, A340165, A340182, A340183, A340293.

Programs

  • Mathematica
    Table[2^(n^2 - 1) * Product[1 + Sin[Pi*j/n]^2 + Sin[Pi*k/n]^2, {j, 1, n}, {k, 1, n}], {n, 0, 10}] // Round

Formula

a(n) = 2^(n^2-1) * Product_{j=1..n, k=1..n} (3 - cos(Pi*j/n)^2 - cos(Pi*k/n)^2).
a(n) = 2^(n^2-1) * Product_{j=1..n, k=1..n} (2-cos(2*Pi*j/n)/2-cos(2*Pi*k/n)/2).
a(n) ~ 2^(n^2-1) * exp(4*c*n^2/Pi^2), where c = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2 + sin(y)^2) dy dx = -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 = A340421 = 1.627008991085721315763766677017604437985734719035793082916212355323520649...
Showing 1-3 of 3 results.

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