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 1-7 of 7 results.

A340052 a(n) = Product_{1<=i

Original entry on oeis.org

1, 1, 5, 91, 5661, 1173821, 801125065, 1786768287095, 12964564030176889, 305121026002697122873, 23243604301636717923421133, 5722586073277932639539150258131, 4548248834078776410469611991220703125
Offset: 0

Views

Author

Seiichi Manyama, Dec 29 2020

Keywords

Links

Crossrefs

Cf. A007726, A065072, A127605.

Programs

  • Mathematica
    Table[2^(n*(n-1)) * Product[Product[Sin[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, j-1}], {j, 2, n}], {n, 0, 15}] // Round (* Vaclav Kotesovec, Dec 30 2020 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(prod(j=2, n, prod(i=1, j-1, 4*sin(i*Pi/(2*n+1))^2+4*sin(j*Pi/(2*n+1))^2)))}

Formula

a(n)^2 = A127605(n)/(2^n * (2*n+1)).
a(n) ~ sqrt(Gamma(1/4)) * exp(G*(2*n+1)^2/(2*Pi)) / (2^(n/2 + 5/4) * Pi^(3/8) * n^(3/4)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Dec 30 2020

A340185 Number of spanning trees in the halved Aztec diamond HOD_n.

Original entry on oeis.org

1, 1, 15, 2639, 5100561, 105518291153, 23067254643457375, 52901008815129395889375, 1266973371422697144030728637409, 315937379766837559600972497421046382689, 818563964325891485548944567913851815851212484079
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2020

Keywords

Comments

*
|
* *---*---*
| | | |
* *---*---* *---*---*---*---*
| | | | | | | | |
*---*---* *---*---*---*---* *---*---*---*---*---*---*
HOD_1 HOD_2 HOD_3
-------------------------------------------------------------
*
|
*---*---*
| | |
*---*---*---*---*
| | | | |
*---*---*---*---*---*---*
| | | | | | |
*---*---*---*---*---*---*---*---*
HOD_4

Links

Crossrefs

Cf. A004003, A007725, A007726, A065072, A127605, A340052, A340176 (halved Aztec diamond HMD_n).

Programs

  • Mathematica
    Table[4^((n-1)*n) * Product[Product[(1 - Cos[j*Pi/(2*n + 1)]^2*Cos[k*Pi/(2*n + 1)]^2), {k, j+1, n}], {j, 1, n}], {n, 0, 12}] // Round (* Vaclav Kotesovec, Jan 03 2021 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(prod(j=1, 2*n, prod(k=j+1, 2*n-j, 4-4*cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))))}
  • PARI
    default(realprecision, 120);
{a(n) = round(4^((n-1)*n)*prod(j=1, n, prod(k=j+1, n, 1-(cos(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))^2)))} \\ Seiichi Manyama, Jan 02 2021
  • Python
    # Using graphillion
from graphillion import GraphSet
def make_HOD(n):
    s = 1
    grids = []
    for i in range(2 * n + 1, 1, -2):
        for j in range(i - 2):
            a, b, c = s + j, s + j + 1, s + i + j
            grids.extend([(a, b), (b, c)])
        grids.append((s + i - 2, s + i - 1))
        s += i
    return grids
def A340185(n):
    if n == 0: return 1
    universe = make_HOD(n)
    GraphSet.set_universe(universe)
    spanning_trees = GraphSet.trees(is_spanning=True)
    return spanning_trees.len()
print([A340185(n) for n in range(7)])

Formula

a(n) = Product_{1<=j
From Seiichi Manyama, Jan 02 2021: (Start)
a(n) = 4^((n-1)*n) * Product_{1<=j
a(n) = A340052(n) * A065072(n) = (1/2^n) * sqrt(A127605(n) * A004003(n) / (2*n+1)). (End)
a(n) ~ sqrt(Gamma(1/4)) * exp(G*(2*n+1)^2/Pi) / (Pi^(3/8) * n^(3/4) * 2^(n + 3/4) * (1 + sqrt(2))^(n + 1/2)), where G is Catalan's constant A006752. - Vaclav Kotesovec, Jan 03 2021

A127606 a(n) = 2^(2*n*n) * Product_{1<=i,j<=n} (cos(i*Pi/(2*n+1))^2 + sin(j*Pi/(2*n+1))^2).

Original entry on oeis.org

1, 4, 176, 79808, 372713728, 17931360207872, 8887976555024756736, 45390122553039546330628096, 2388340820825093234015277927170048, 1294826675280341699389150405743029631844352
Offset: 0

Views

Author

Miklos Kristof, Apr 03 2007

Keywords

Comments

a(n)/4^n is an integer. - Seiichi Manyama, Dec 31 2020

Links

Crossrefs

Cf. A004003, A127605.

Programs

  • Maple
    for n from 0 to 12 do a[n]:=2^(2*n*n)*product(product(cos(i*Pi/(2*n+1))^2+ sin(j*Pi/(2*n+1))^2,j=1..n),i=1..n) od: seq(round(evalf(a[n],300)),n=0..12);
  • Mathematica
    Table[2^(2*n^2) * Product[Product[Cos[i*Pi/(2*n + 1)]^2 + Sin[j*Pi/(2*n + 1)]^2, {i, 1, n}], {j, 1, n}], {n, 0, 15}] // Round (* Vaclav Kotesovec, Mar 18 2023 *)
  • PARI
    default(realprecision, 120);
{a(n) = round(prod(i=1, n, prod(j=1, n, 4*cos(i*Pi/(2*n+1))^2+4*sin(j*Pi/(2*n+1))^2)))} \\ Seiichi Manyama, Dec 31 2020
  • PARI
    {a(n) = sqrtint(4^n*polresultant(polchebyshev(2*n+1, 1, I*x/2), polchebyshev(2*n, 2, x/2)))} \\ Seiichi Manyama, Jan 09 2021
  • Python
    from math import isqrt
from sympy.abc import x
from sympy import resultant, chebyshevt, chebyshevu, I
def A127606(n): return isqrt(resultant(chebyshevt((n<<1)+1,I*x/2),chebyshevu(n<<1,x/2)))<Chai Wah Wu, Nov 07 2023

Formula

a(n) = 2^n * sqrt(Resultant(T_{2*n+1}(i*x/2), U_{2*n}(x/2))), where T_n(x) is a Chebyshev polynomial of the first kind, U_n(x) is a Chebyshev polynomial of the second kind and i = sqrt(-1). - Seiichi Manyama, Jan 09 2021
a(n) ~ 2^(1/8) * exp(G*(2*n + 1)^2/Pi) / (1 + sqrt(2))^(n + 1/2), where G is Catalan's constant A006752. - Vaclav Kotesovec, Mar 18 2023

A340475 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals, where T(n,k) = Product_{a=1..n} Product_{b=1..k} (4*sin(a*Pi/(2*n+1))^2 + 4*sin(b*Pi/(2*k+1))^2).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 29, 29, 1, 1, 139, 500, 139, 1, 1, 666, 8329, 8329, 666, 1, 1, 3191, 138301, 463736, 138301, 3191, 1, 1, 15289, 2295701, 25543057, 25543057, 2295701, 15289, 1, 1, 73254, 38105729, 1404312491, 4614756624, 1404312491, 38105729, 73254, 1
Offset: 0

Views

Author

Seiichi Manyama, Jan 09 2021

Keywords

Examples

			Square array begins:
  1,   1,      1,        1,          1, ...
  1,   6,     29,      139,        666, ...
  1,  29,    500,     8329,     138301, ...
  1, 139,   8329,   463736,   25543057, ...
  1, 666, 138301, 25543057, 4614756624, ...

Crossrefs

Rows and columns 0..1 give A000012, A030221.
Main diagonal gives A127605.
Cf. A116469, A187617, A340476.

Programs

  • PARI
    default(realprecision, 120);
{T(n, k) = round(prod(a=1, n, prod(b=1, k, 4*sin(a*Pi/(2*n+1))^2+4*sin(b*Pi/(2*k+1))^2)))}

Formula

T(n,k) = T(k,n).

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).

Original entry on oeis.org

1, 9, 7486875, 14334918272193811385583, 1483160703050490588200236172057973908184332257091136
Offset: 0

Views

Author

Seiichi Manyama, Dec 31 2020

Keywords

Comments

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

Crossrefs

Cf. A124647, A127605, A340182, A340183.

Programs

  • Mathematica
    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 *)
  • PARI
    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))))}

Formula

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

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...

A340168 Decimal expansion of a constant related to the asymptotics of A004003.

Original entry on oeis.org

1, 1, 0, 8, 8, 6, 2, 2, 5, 8, 7, 8, 0, 7, 6, 7, 5, 1, 3, 2, 7, 6, 9, 5, 1, 1, 6, 2, 1, 3, 0, 8, 1, 9, 2, 9, 2, 6, 4, 5, 2, 6, 6, 1, 2, 6, 9, 6, 3, 5, 6, 9, 2, 2, 4, 3, 6, 2, 9, 4, 3, 1, 4, 1, 8, 4, 4, 7, 3, 5, 5, 6, 5, 3, 0, 9, 3, 4, 8, 6, 6, 3, 2, 1, 3, 4, 3, 9, 7, 1, 4, 6, 7, 5, 0, 7, 9, 0, 1, 5, 5, 7, 4, 0, 5
Offset: 1

Views

Author

Vaclav Kotesovec, Dec 30 2020

Keywords

Examples

			1.1088622587807675132769511621308192926452661269635692243629431418447355653...

Crossrefs

Cf. A004003, A007341, A007726, A065072, A127605, A340052.

Programs

  • Mathematica
    RealDigits[2*E^(Catalan/Pi)/(1 + Sqrt[2]), 10, 110][[1]]

Formula

Equals lim_{n->infinity} A004003(n) / ((sqrt(2)-1)^(2*n) * exp(4*G*n*(n+1)/Pi)), where G is the Catalan's constant A006752.
Equals 2*exp(G/Pi) / (1 + sqrt(2)), where G is Catalan's constant A006752.
Showing 1-7 of 7 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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