A340322 -id:A340322 - cp's OEIS frontend

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

1, 3, 61731, 220157391087140625, 3109768877542258728107559478225309328087616

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

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

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 *)

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))))}

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)

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)).

A071763 Number of spanning trees in n X n X n grid.

Original entry on oeis.org

1, 384, 8193540096000, 172685928902844729688524604506636288, 77746347057132811936046563068332100246216273086593103906734080000000000000

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Jun 04 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], 2000.
Eric Weisstein's World of Mathematics, Grid Graph
Eric Weisstein's World of Mathematics, Spanning Tree

Cf. A007341, A340182.

Table[2^(n^3 - 1)/n^3 Product[Piecewise[{{1, i == j == k == 0}}, 3 - Cos[Pi i/n] - Cos[Pi j/n] - Cos[Pi k/n]], {i, 0, n - 1}, {j, 0, n - 1}, {k, 0, n - 1}], {n, 12}] // Round

a(n) = 2^(n^3-1) / n^3 * Product_{n1=0..n-1 n2=0..n-1 n3=0..n-1} (3- cos(Pi*n1/n) - cos(Pi*n2/n) - cos(Pi*n3/n) ) where n1, n2, n3 are not all 0.

Limit_{n->infinity} a(n)^(1/n^3) = exp(8 * A340322 / Pi^3) = 5.330202889205167421134597996649659520108446730592285502966091902480522584119... - Vaclav Kotesovec, Jan 05 2021

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.

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

Vaclav Kotesovec, Jan 07 2021

			2.551698806403924360993561678605629314325436926549295727591221339383517201757...

Cf. A067518, A340322, A340350, A340421.

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]]

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.

A340183 a(n) = Product_{1<=j,k,m<=n-1} (4sin(jPi/(2n))^2 + 4sin(kPi/(2n))^2 + 4sin(mPi/(2*n))^2).

Original entry on oeis.org

1, 6, 1157625, 170875128460147163136, 448524809573174705684873233798538664686384705625

Offset: 1

Seiichi Manyama, Dec 31 2020

nonn

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

Cf. A007341, A124647, A340181, A340182.

Round[Table[2^((n-1)^3)* Product[3 - Cos[j*Pi/n] - Cos[k*Pi/n] - Cos[m*Pi/n], {j, 1, n-1}, {k, 1, n-1}, {m, 1, n-1}], {n, 1, 5}]] (* Vaclav Kotesovec, Jan 04 2021 *)

default(realprecision, 500);
{a(n) = round(prod(j=1, n-1, prod(k=1, n-1, prod(m=1, n-1, 4*sin(j*Pi/(2*n))^2+4*sin(k*Pi/(2*n))^2+4*sin(m*Pi/(2*n))^2))))}

a(n) = Product_{1<=i,j,k<=n-1} (4*f(i*Pi/(2*n))^2 + 4*g(j*Pi/(2*n))^2 + 4*h(k*Pi/(2*n))^2), where f(x), g(x) and h(x) are sin(x) or cos(x).

Limit_{n->infinity} a(n)^(1/n^3) = exp(8*A340322/Pi^3). - Vaclav Kotesovec, Jan 05 2021

A340181 a(n) = Product_{1<=j,k,m<=n} (4sin(jPi/(2n+1))^2 + 4sin(kPi/(2n+1))^2 + 4sin(mPi/(2*n+1))^2).

Original entry on oeis.org

1, 9, 7486875, 14334918272193811385583, 1483160703050490588200236172057973908184332257091136

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

(a(n)/((2n + 1)*3^n))^(1/3) is an integer.

Cf. A124647, A127605, A340182, A340183.

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}]] (* Vaclav Kotesovec, Jan 04 2021 *)

default(realprecision, 500);
{a(n) = round(prod(j=1, n, prod(k=1, n, prod(m=1, n, 4*sin(j*Pi/(2*n+1))^2+4*sin(k*Pi/(2*n+1))^2+4*sin(m*Pi/(2*n+1))^2))))}

Limit_{n->infinity} a(n)^(1/n^3) = exp(8*A340322/Pi^3). - Vaclav Kotesovec, Jan 05 2021

cp's OEIS Frontend

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).

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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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A071763 Number of spanning trees in n X n X n grid.

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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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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.

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A340183 a(n) = Product_{1<=j,k,m<=n-1} (4*sin(j*Pi/(2*n))^2 + 4*sin(k*Pi/(2*n))^2 + 4*sin(m*Pi/(2*n))^2).

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A340181 a(n) = Product_{1<=j,k,m<=n} (4*sin(j*Pi/(2*n+1))^2 + 4*sin(k*Pi/(2*n+1))^2 + 4*sin(m*Pi/(2*n+1))^2).

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

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

A340183 a(n) = Product_{1<=j,k,m<=n-1} (4sin(jPi/(2n))^2 + 4sin(kPi/(2n))^2 + 4sin(mPi/(2*n))^2).

A340181 a(n) = Product_{1<=j,k,m<=n} (4sin(jPi/(2n+1))^2 + 4sin(kPi/(2n+1))^2 + 4sin(mPi/(2*n+1))^2).