A340422 -id:A340422 - cp's OEIS frontend

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

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

Vaclav Kotesovec, Jan 04 2021

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.

			6.485696465218497693708581372103315764152266325617976316831738842452555238784...

Vaclav Kotesovec, Table of n, a(n) for n = 1..200

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

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));

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

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.

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

Vaclav Kotesovec, Jan 05 2021

			0.49531661869212336430296504041161047588717884176797451824647459341123774...

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

evalf(Pi * Integrate(log((1 + sqrt(1 + sin(x)^2))/2), x = 0..Pi/2), 120);

RealDigits[N[Pi*Integrate[Log[(1 + Sqrt[1 + Sin[x]^2])/2], {x, 0, Pi/2}], 100]][[1]]

Pi * intnum(x = 0, Pi/2, log((1 + sqrt(1 + sin(x)^2))/2))

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)).

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

Vaclav Kotesovec, Jan 07 2021

			1.627008991085721315763766677017604437985734719035793082916212355323520769...

Cf. A340396, A340322, A340350, A340422.

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]]

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.

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

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

nonn

W.-J. Tzeng and F. Y. Wu, Spanning Trees on Hypercubic Lattices and Non-orientable Surfaces, arXiv:cond-mat/0001408 [cond-mat.stat-mech], 2002.

Cf. A071763, A007341, A340396.

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 *)

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

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

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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.

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A340350 Decimal expansion of Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2*sin(y)^2) dy dx.

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

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

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