A334088
a(n) = sqrt(Resultant(T(2*n,x/2), T(2*n,i*x/2))), where T(n,x) is a Chebyshev polynomial of the first kind and i = sqrt(-1).
Original entry on oeis.org
1, 1, 8, 676, 591872, 5347119376, 497996601804800, 477995151754478453824, 4727827717838439286122217472, 481856411624794348153802518369517824, 506033683217425527860454091268429289861152768
Offset: 0
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Table[Sqrt[Resultant[ChebyshevT[2*n, x/2], ChebyshevT[2*n, I*x/2], x]], {n, 0, 12}] (* Vaclav Kotesovec, Apr 14 2020 *)
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{a(n) = sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))}
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from math import isqrt
from sympy.abc import x
from sympy import resultant, chebyshevt, I
def A334088(n): return isqrt(resultant(chebyshevt(n<<1,x/2),chebyshevt(n<<1,I*x/2))) if n else 1 # Chai Wah Wu, Nov 07 2023
A340176
Number of spanning trees in the halved Aztec diamond HMD_n.
Original entry on oeis.org
1, 1, 4, 208, 121856, 772189440, 51989627289600, 36837279603595907072, 273129993621426778551615488, 21114078836429317912110529666154496, 16975032309392309949804839529585109326888960
Offset: 0
a(2) = 4;
* * *---* *---* *---*
| | | | | |
*---*---*---* *---*---*---* *---*---*---* *---* *---*
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default(realprecision, 120);
{a(n) = round(prod(j=1, 2*n-1, prod(k=j+1, 2*n-1-j, 4-4*cos(j*Pi/(2*n))*cos(k*Pi/(2*n)))))}
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{a007341(n) = polresultant(polchebyshev(n-1, 2, x/2), polchebyshev(n-1, 2, (4-x)/2))};
{a334088(n) = sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))};
{a(n) = if(n==0, 1, sqrtint(a007341(n)*a334088(n)/n))}
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default(realprecision, 120);
{a(n) = if(n==0, 1, round(4^((n-1)^2)*prod(j=1, n-1, prod(k=j+1, n-1, 1-(cos(j*Pi/(2*n))*cos(k*Pi/(2*n)))^2))))} \\ Seiichi Manyama, Jan 02 2021
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# Using graphillion
from graphillion import GraphSet
def make_HMD(n):
s = 1
grids = []
for i in range(2 * n, 0, -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 A340176(n):
if n == 0: return 1
universe = make_HMD(n)
GraphSet.set_universe(universe)
spanning_trees = GraphSet.trees(is_spanning=True)
return spanning_trees.len()
print([A340176(n) for n in range(7)])
A340295
a(n) = 4^(2*n^2) * Product_{1<=j,k<=n} (1 - sin(j*Pi/(2*n+1))^2 * cos(k*Pi/(2*n+1))^2).
Original entry on oeis.org
1, 13, 18281, 2732887529, 43384923739812577, 73125714588602035608260981, 13085551252412040683513520733767180041, 248596840858215958581954513797323868183183928594833
Offset: 0
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Table[Resultant[ChebyshevT[4*n+2, x/2], ChebyshevT[4*n+2, I*x/2], x]^(1/4) / 2^n, {n, 0, 10}] (* Vaclav Kotesovec, Jan 04 2021 *)
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default(realprecision, 120);
{a(n) = round(4^(2*n^2)*prod(j=1, n, prod(k=1, n, 1-(sin(j*Pi/(2*n+1))*cos(k*Pi/(2*n+1)))^2)))}
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{a(n) = sqrtint(sqrtint(polresultant(polchebyshev(4*n+2, 1, x/2), polchebyshev(4*n+2, 1, I*x/2))))/2^n}
Showing 1-3 of 3 results.
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