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 1-4 of 4 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

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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.

A340182 a(n) = Product_{1<=j,k,m<=n} (4*cos(j*Pi/(2*n+1))^2 + 4*cos(k*Pi/(2*n+1))^2 + 4*cos(m*Pi/(2*n+1))^2).

Original entry on oeis.org

1, 3, 61731, 220157391087140625, 3109768877542258728107559478225309328087616
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2020

Keywords

Comments

(a(n)/3^n)^(1/3) is an integer.

Crossrefs

Cf. A004003, A071763, A340181, A340183.

Programs

  • Mathematica
    Round[Table[4^(n^3) * Product[Cos[j*Pi/(2*n + 1)]^2 + Cos[k*Pi/(2*n + 1)]^2 + Cos[m*Pi/(2*n + 1)]^2, {j, 1, n}, {k, 1, n}, {m, 1, n}], {n, 0, 5}]] (* or *)
Round[Table[2^(n^3) * Product[3 + Cos[2*j*Pi/(2*n + 1)] + Cos[2*k*Pi/(2*n + 1)] + Cos[2*m*Pi/(2*n + 1)], {j, 1, n}, {k, 1, n}, {m, 1, n}], {n, 0, 5}]] (* or *)
Round[Table[Product[u = Sqrt[Cos[j*Pi/(2*n + 1)]^2 + Cos[k*Pi/(2*n + 1)]^2]; (((u + Sqrt[1 + u^2])^(2*n + 1) - (u - Sqrt[1 + u^2])^(2*n + 1))/(2*Sqrt[1 + u^2])), {j, 1, n}, {k, 1, n}], {n, 0, 5}]] (* Vaclav Kotesovec, Jan 04 2021 *)
  • PARI
    default(realprecision, 500);
{a(n) = round(prod(j=1, n, prod(k=1, n, prod(m=1, n, 4*cos(j*Pi/(2*n+1))^2+4*cos(k*Pi/(2*n+1))^2+4*cos(m*Pi/(2*n+1))^2))))}

Formula

From Vaclav Kotesovec, Jan 04 2021: (Start)
a(n) ~ c * d^n * s^(n^2) * r^(n^3), where
r = exp(8*A340322/Pi^3) = exp((8/Pi^3) * 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) dx dy dz) = 5.3302028892051674211345979966496595201084467305922855029660919024805225841...
s = 0.57208914727550556482486188829703578692890272003698306852389010626941042...
d = 0.91012013388841787275362130594290903074302493828277326742531159...
c = 1.057086458532774496412062406469810663638243576302292119... (End)

A067518 Number of spanning trees in n X n X 2 grid.

Original entry on oeis.org

1, 384, 49766400, 2200248344641536, 32699232783861202944000000, 161655300770215803222365206216704000000, 264237966861625003904099008804894577790426446838104064
Offset: 1

Views

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jun 08 2002

Keywords

Links

Crossrefs

Cf. A071763, A007341, A340396.

Programs

  • Mathematica
    a[n_] := 2^(2*n^2 - 2)/n^2*Product[If[n1+n2+n3 > 0, 3 - Cos[Pi*n1/n] - Cos[Pi*n2/n] - Cos[Pi*n3/2], 1], {n1, 0, n-1}, {n2, 0, n-1}, {n3, 0, 1}];
Table[a[n] // Round, {n, 1, 7}] (* Jean-François Alcover, Feb 18 2019 *)

Formula

a(n) = 2^(2*n^2-2) / n^2 * Product_{n1=0..n-1, n2=0..n-1, n3=0..1, n1+n2+n3>0} (3 - cos(Pi*n1/n) - cos(Pi*n2/n) - cos(Pi*n3/2)).
a(n) ~ c * d^n * 2^(n^2) * exp(4*n^2*(G/Pi + m/Pi^2)) / sqrt(n), where m = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + 2*sin(x)^2 + 2*sin(y)^2) dy dx = A340422 = 2.5516988064039243609935616786056293143254369265492957275912213393835172..., d = (2*sqrt(2)-3)*(2+sqrt(3))*(sqrt(15)-4) = 0.08133113706589390743806107..., c = 5^(1/4) * Gamma(1/4) / (sqrt(3) * (2*Pi)^(3/4)) = 0.788729432659299631982768... and G is Catalan's constant A006752. Equivalently, m = 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. - Vaclav Kotesovec, Jan 06 2021, updated Mar 17 2024

Extensions

More terms from André Pönitz (poenitz(AT)htwm.de), Jun 11 2003

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 1-4 of 4 results.

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