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.

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

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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.

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...

A338832 Number of spanning trees in the k_1 X ... X k_j grid graph, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.

Original entry on oeis.org

1, 1, 1, 4, 1, 15, 1, 384, 192, 56, 1, 31500, 1, 209, 2415, 42467328, 1, 49766400, 1, 2558976, 30305, 780, 1, 3500658000000, 100352, 2911, 8193540096000, 207746836, 1, 76752081000, 1, 20776019874734407680, 380160, 10864, 4140081, 242716067758080000000, 1
Offset: 1

Views

Author

Pontus von Brömssen, Nov 11 2020

Keywords

Comments

a(n) > 1 precisely when n is composite.

Examples

			The partition (2, 2, 1) has Heinz number 18 and the 3 X 3 X 2 grid graph has a(18) = 49766400 spanning trees.

Links

Crossrefs

2 X n grid: A001353(n) = a(2*prime(n-1))
3 X n grid: A006238(n) = a(3*prime(n-1))
4 X n grid: A003696(n) = a(5*prime(n-1))
5 X n grid: A003779(n) = a(7*prime(n-1))
6 X n grid: A139400(n) = a(11*prime(n-1))
7 X n grid: A334002(n) = a(13*prime(n-1))
8 X n grid: A334003(n) = a(17*prime(n-1))
9 X n grid: A334004(n) = a(19*prime(n-1))
10 X n grid: A334005(n) = a(23*prime(n-1))
n X n grid: A007341(n) = a(prime(n-1)^2)
m X n grid: A116469(m,n) = a(prime(m-1)*prime(n-1))
2 X 2 X n grid: A003753(n) = a(4*prime(n-1))
2 X n X n grid: A067518(n) = a(2*prime(n-1)^2)
n X n X n grid: A071763(n) = a(prime(n-1)^3)
2 X ... X 2 grid: A006237(n) = a(2^n)

Formula

a(n) = Product_{n_1=0..k_1-1, ..., n_j=0..k_j-1; not all n_i=0} Sum_{i=1..j} (2*(1 - cos(n_i*Pi/k_i))) / Product_{i=1..j} k_i, where (k_1 - 1, ..., k_j - 1) is the partition with Heinz number n.
Showing 1-3 of 3 results.

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