A340167 -id:A340167 - cp's OEIS frontend

A340166 a(n) = 4^(2(n-1)^2) Product_{1<=i,j<=n-1} (1 - sin(iPi/(2n))^2 * sin(jPi/(2n))^2).

1, 12, 17745, 2958176256, 54090331699622625, 107181043200192494332800000, 22868509031094388112997259982567521313, 523389340935243821042846225254323436248483571433472

Offset: 1

Seiichi Manyama, Dec 30 2020

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..30
D. E. Knuth, Aztec Diamonds, Checkerboard Graphs, and Spanning Trees, arXiv:math/9501234 [math.CO], 1995; J. Alg. Combinatorics 6 (1997), 253-257.

Cf. A007725, A340139, A340167.

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 *)

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)))}

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) ~ Gamma(1/4) * exp(8*G*n^2/Pi) / (Pi^(3/4) * sqrt(n) * 2^(6*n - 2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 05 2021

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)).

A340165 a(n) = 4^((n-2)(n-1)) Product_{1<=i

Original entry on oeis.org

1, 1, 19, 7056, 51251277, 7280323311888, 20225477546584790663, 1098876823994281426921193472, 1167619533875635661974056722756222809, 24263631353490502503207804571072304043237024000

Offset: 1

Seiichi Manyama, Dec 30 2020

nonn

Cf. A340139, A340167.

Table[4^((n-2)*(n-1)) * Product[Product[1 + Sin[i*Pi/(2*n)]^2 * Sin[j*Pi/(2*n)]^2, {i, 1, j-1}], {j, 2, n-1}], {n, 1, 12}] // Round (* Vaclav Kotesovec, Dec 31 2020 *)

default(realprecision, 120);
{a(n) = round(4^((n-2)*(n-1))*prod(j=2, n-1, prod(i=1, j-1, 1+(sin(i*Pi/(2*n))*sin(j*Pi/(2*n)))^2)))}

a(n) = 4^((n-2)*(n-1)) * Product_{1<=i

a(n) ~ 2^(2*n^2 - 3*n + 35/8) * (1 - sqrt(2*sqrt(2)-2))^n * exp(2*A340350*n^2/Pi^2). - Vaclav Kotesovec, Jan 05 2021

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cp's OEIS Frontend

A340166 a(n) = 4^(2(n-1)^2) Product_{1<=i,j<=n-1} (1 - sin(iPi/(2n))^2 * sin(jPi/(2n))^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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A340165 a(n) = 4^((n-2)(n-1)) Product_{1<=i

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cp's OEIS Frontend

A340166 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).

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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.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A340165 a(n) = 4^((n-2)*(n-1)) * Product_{1<=i

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Author

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Programs

Mathematica

PARI

Formula

A340166 a(n) = 4^(2(n-1)^2) Product_{1<=i,j<=n-1} (1 - sin(iPi/(2n))^2 * sin(jPi/(2n))^2).

A340165 a(n) = 4^((n-2)(n-1)) Product_{1<=i