A004003 -id:A004003 - cp's OEIS frontend

1, 1, 15, 2639, 5100561, 105518291153, 23067254643457375, 52901008815129395889375, 1266973371422697144030728637409, 315937379766837559600972497421046382689, 818563964325891485548944567913851815851212484079

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..40
Mihai Ciucu, Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole, arXiv:0710.4500 [math.CO], 2007. See Corollary 3.7.

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

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 *)

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)))))}

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

# 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)])

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

A348452 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area.

Original entry on oeis.org

1, 1, 2, 0, 1, 1, 0, 10, 0, 0, 0, 0, 0, 1, 1, 70, 0, 117, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 80518, 264500, 442791, 0, 451206, 0, 0, 178939, 0, 0, 80092, 0, 0, 0, 0, 0, 6728, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

N. J. A. Sloane, Oct 27 2021

The board has n^2 squares. The colors do not matter. T(n,k) is zero unless k divides n^2. The tiles are rook-connected polygons made from n^2/k squares.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
A348453 (the main entry for this problem) displays the same data in a more compact way (by omitting the zero entries from each row).
The data is taken from A004003, A172477, and Schutzman & MGGG (2018).

			The first seven rows of the triangle are:
1,
1, 2, 0, 1,
1, 0, 10, 0, 0, 0, 0, 0, 1,
1, 70, 0, 117, 0, 0, 0, 36, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 4006, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 80518, 264500, 442791, 0, 451206, 0, 0, 178939, 0, 0, 80092, 0, 0, 0, 0, 0, 6728, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
1, 0, 0, 0, 0, 0, 158753814, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
...
The domino is the only polyomino of area 2, and the 36 ways to tile a 4 X 4 square with dominoes are shown in one of the links.

Moon Duchin, Graphs, Geometry and Gerrymandering”, Talk at Celebration of Mind Conference, Oct 23 2021.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Zach Schutzman and MGGG, The Known Sizes of Grid Metagraphs, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.
N. J. A. Sloane, Illustration for T(3,3) = 10
N. J. A. Sloane, Illustration for T(4,2) = 70 [Labels give code, B = length of internal boundary, C = number of internal corners, G = group order, # = number of this type. Note that (B,C) determines the type]
N. J. A. Sloane, Illustration for T(4,8) = 36 [Slide from an old talk of mine]
N. J. A. Sloane, The On-Line Encyclopedia of Integer Sequences: An illustrated guide with many unsolved problems, Guest Lecture given in Doron Zeilberger's Experimental Mathematics Math640 Class, Rutgers University, Spring Semester, Apr 28 2022: Slides; Slides (an alternative source).

Cf. A348453. A348454 and A348455 are similar triangles with the data in each row reversed. The row sums are in A348789.

See also A004003, A099390, A172477, A348456.

A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.

More than the usual number of terms are given, in order to show the first seven rows.

A334088 a(n) = sqrt(Resultant(T(2n,x/2), T(2n,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

Seiichi Manyama, Apr 14 2020

nonn

Wikipedia, Chebyshev polynomials
Wikipedia, Resultant

Cf. A004003, A334089.

Table[Sqrt[Resultant[ChebyshevT[2*n, x/2], ChebyshevT[2*n, I*x/2], x]], {n, 0, 12}] (* Vaclav Kotesovec, Apr 14 2020 *)

{a(n) = sqrtint(polresultant(polchebyshev(2*n, 1, x/2), polchebyshev(2*n, 1, I*x/2)))}

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

a(n) ~ exp(4*G*n^2/Pi) / 2^(2*n - 1/4), where G is Catalan's constant A006752. - Vaclav Kotesovec, Apr 14 2020

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).

Original entry on oeis.org

1, 3, 61731, 220157391087140625, 3109768877542258728107559478225309328087616

Offset: 0

Seiichi Manyama, Dec 31 2020

nonn

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

Cf. A004003, A071763, A340181, A340183.

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)

A348453 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.

Original entry on oeis.org

1, 1, 2, 1, 1, 10, 1, 1, 70, 117, 36, 1, 1, 4006, 1, 1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1, 1, 158753814, 1, 7157114189

Offset: 1

N. J. A. Sloane, Oct 27 2021

The board has n^2 squares. The colors do not matter. The tiles are rook-connected polygons made from n^2/d_k squares.
This is the "labeled" version of the problem. Symmetries of the square are not taken into account. Rotations and reflections count as different.
A348452 displays the same data in a less compact way. The present triangle is obtained by omitting the zero entries from A348452.
The data is taken from A004003, A172477, A348456, and Schutzman & MGGG (2018).
T(8,2) = 7157114189 (see A348456). T(8,3) is presently unknown.

			The first eight rows of the triangle are:
  1,
  1, 2, 1,
  1, 10, 1,
  1, 70, 117, 36, 1,
  1, 4006, 1,
  1, 80518, 264500, 442791, 451206, 178939, 80092, 6728, 1,
  1, 158753814, 1,
  1, 7157114189, ?, 187497290034, ?, ?, 1,
  ...
The corresponding divisors d_k are:
  1,
  1, 2, 4,
  1, 3, 9,
  1, 2, 4, 8, 16,
  1, 5, 25,
  ...
The domino is the only polyomino of area 2, and the 36 ways to tile a 4 X 4 square with dominoes are shown in one of the links.

Moon Duchin, Graphs, Geometry and Gerrymandering”, Talk at Celebration of Mind Conference, Oct 23 2021.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
P. W. Kasteleyn, Dimer statistics and phase transitions, J. Mathematical Phys. 4 1963 287-293. MR0153427 (27 #3394).
Zach Schutzman and MGGG, The Known Sizes of Grid Metagraphs, Metric Geometry and Gerrymandering Group (MGGG), Boston, Oct 01 2018.
N. J. A. Sloane, Illustration for T(3,2) = 10
N. J. A. Sloane, Illustration for T(4,2) = 70 [Labels give code, B = length of internal boundary, C = number of internal corners, G = group order, # = number of this type. Note that (B,C) determines the type]
N. J. A. Sloane, Illustration for T(4,4) = 36 [Slide from an old talk of mine]
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 21.

Cf. A348452. A348454 and A348455 are similar triangles with the data in each row reversed.
See also A004003, A099390, A172477, A348456.
Cf. A048691 (row lengths).

A formula for T(n, n^2/2) was found by Kastelyn (see A004003 and A099390). T(n,n) is studied in A172477.

T(8,2) added May 04 2022 (see A348456) - N. J. A. Sloane, May 05 2022

A360256 Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.

Original entry on oeis.org

1, 1, 33, 513, 14409, 693025, 50447161

Offset: 1

Scott R. Shannon, Feb 17 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct height X width dimensions means that, for example, a 1 X 3 rectangle can be used twice, once in a horizontal (1 X 3) and once in a vertical (3 X 1) direction.

			a(1) = 1 as the only way to tile a 1 X 1 square is with a square with dimensions 1 X 1.
a(2) = 1 as the only way to tile a 2 X 2 square is with a square with dimensions 2 X 2.
a(3) = 33. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
  |   |       |   |   |       |   |       |   |   |           |   |   |       |
  +---+---+---+   +---+---+---+   +---+---+   +   +---+---+---+   +---+---+---+
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +   +       +   +           +   +       +   +   +           +   +       +   +
  |   |       |   |           |   |       |   |   |           |   |       |   |
  +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+   +---+---+---+
.
The first tiling can occur in 4 different ways, the second in 8 different ways, the third in 8 different ways, the fourth in 4 different ways and the fifth in 8 different ways. There is also the single 3 X 3 rectangle. This gives 33 ways in total.

Cf. A360725 (oblongs), A360499, A360498, A182275, A004003, A099390, A065072.

A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

Original entry on oeis.org

0, 0, 4, 12, 256, 3620, 87216, 2444084, 87181220

Offset: 1

Scott R. Shannon, Feb 09 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 oblong can only be used once, regardless of if it lies horizontally or vertically.

			a(1) = 0 as no distinct oblongs can tile a square with dimensions 1 x 1.
a(2) = 0 as no distinct oblongs can tile a square with dimensions 2 x 2.
a(3) = 4. There is one tiling, excluding those equivalent by symmetry:
.
  +---+---+---+
  |           |
  +---+---+---+
  |           |
  +           +
  |           |
  +---+---+---+
.
This tiling can occur in 4 different ways, giving 4 ways in total.
a(4) = 12. The possible tilings, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+   +---+---+---+---+
  |   |           |   |               |
  +   +           +   +---+---+---+---+
  |   |           |   |               |
  +---+---+---+---+   +               +
  |               |   |               |
  +               +   +               +
  |               |   |               |
  +---+---+---+---+   +---+---+---+---+
.
The first tiling can occur in 8 different way and the second in 4 different ways, giving 12 ways in total.

Cf. A360499 (rectangles), A004003, A099390, A065072, A233320, A230031.

A239273 Number of domicule tilings of a 2n X 2n square grid.

Original entry on oeis.org

1, 3, 280, 3037561, 3263262629905, 326207195516663381931, 3011882198082438957330143630563, 2565014347691062208319404612723752103028288, 201442620359313683494245316355883565275531844406384955392, 1458834332808489549111708247664894524221330758005874053074138540424018259

Offset: 0

Alois P. Heinz, Mar 13 2014

nonn

A domicule is either a domino or it is formed by the union of two neighboring unit squares connected via their corners. In a tiling the connections of two domicules are allowed to cross each other.

Number of perfect matchings in the 2n X 2n kings graph. - Andrew Howroyd, Apr 07 2016

			a(1) = 3:
  +---+   +---+   +---+
  |o o|   |o o|   |o-o|
  || ||   | X |   |   |
  |o o|   |o o|   |o-o|
  +---+   +---+   +---+.
a(2) = 280:
  +-------+ +-------+ +-------+ +-------+ +-------+
  |o o o-o| |o o o-o| |o-o o-o| |o o o o| |o o-o o|
  | X     | | X     | |       | | X  | || | \   / |
  |o o o o| |o o o o| |o o o o| |o o o o| |o o o o|
  |   /  || |   / / | ||  X  || |       | ||     ||
  |o o o o| |o o o o| |o o o o| |o-o o o| |o o o o|
  ||    \ | ||     || |       | |     X | | / /   |
  |o o-o o| |o o-o o| |o-o o-o| |o-o o o| |o o o-o|
  +-------+ +-------+ +-------+ +-------+ +-------+ ...

Eric Weisstein's World of Mathematics, King Graph
Wikipedia, King's graph

Even bisection of main diagonal of A239264.

Cf. A004003, A243510, A243424, A220638.

b[n_, l_List] := b[n, l] = Module[{d = Length[l]/2, f = False, k}, Which[n == 0, 1, l[[1 ;; d]] == Array[f &, d], b[n - 1, Join[l[[d + 1 ;; 2*d]], Array[True &, d]]], True, For[k = 1, ! l[[k]], k++]; If[k < d && n > 1 && l[[k + d + 1]], b[n, ReplacePart[l, {k -> f, k + d + 1 -> f}]], 0] + If[k > 1 && n > 1 && l[[k + d - 1]], b[n, ReplacePart[l, {k -> f, k + d - 1 -> f}]], 0] + If[n > 1 && l[[k + d]], b[n, ReplacePart[l, {k -> f, k + d -> f}]], 0] + If[k < d && l[[k + 1]], b[n, ReplacePart[l, {k -> f, k + 1 -> f}]], 0]]];
A[n_, k_] := If[Mod[n*k, 2]>0, 0, If[k>n, A[k, n], b[n, Array[True&, k*2]]]];
a[n_] := A[2n, 2n];
Table[Print[n]; a[n], {n, 0, 7}] (* Jean-François Alcover, Sep 16 2019, after Alois P. Heinz in A239264 *)

a(n) = A239264(2n,2n).

a(8) from Alois P. Heinz, Sep 30 2014

a(9) from Alois P. Heinz, Nov 23 2018

A038758 Number of ways of covering a 2n X 2n lattice by 2n^2 dominoes with exactly 4 horizontal (or vertical) dominoes.

Original entry on oeis.org

16, 281, 1785, 7175, 22015, 56406, 126966, 259170, 490050, 871255, 1472471, 2385201, 3726905, 5645500, 8324220, 11986836, 16903236, 23395365, 31843525, 42693035, 56461251, 73744946, 95228050, 121689750, 154012950, 193193091

Offset: 2

Yong Kong (ykong(AT)curagen.com), May 06 2000

			a(3) = 281 because we have 281 ways to cover a 4 X 4 lattice with exactly 4 horizontal dominoes and exactly 14 vertical dominoes.

Vincenzo Librandi, Table of n, a(n) for n = 2..1000
M. E. Fisher, Statistical mechanics of dimers on a plane lattice, Physical Review, 124 (1961), 1664-1672.
P. W. Kasteleyn, The Statistics of Dimers on a Lattice, Physica, 27 (1961), 1209-1225.
Index entries for sequences related to dominoes
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A004003, A002414, A054344.

[(1/24)*n*(n-1)*(n+1)*(12*n^3-11*n-10): n in [2..30]]; // Vincenzo Librandi, Oct 22 2013

CoefficientList[Series[(16 + 169 x + 154 x^2 + 21 x^3)/(1 - x)^7, {x, 0, 30}], x] (* Vincenzo Librandi, Oct 22 2013 *)

a(n) = (1/24)*n*(n-1)*(n+1)*(12*n^3-11*n-10).

G.f.: x^2*(16+169*x+154*x^2+21*x^3)/(1-x)^7. [Colin Barker, Jun 26 2012]

More terms from James Sellers, May 10 2000

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cp's OEIS Frontend

A335586 Number of domino tilings of a 2n X 2n toroidal grid.

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

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A348452 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n^2) is the number of ways to tile an n X n chessboard with k rook-connected polyominoes of equal area.

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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).

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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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A348453 Irregular triangle read by rows: T(n,k) (n >= 1, 1 <= k <= number of divisors of n^2) is the number of ways to tile an n X n chessboard with d_k rook-connected polyominoes of equal area, where d_k is the k-th divisor of n^2.

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A360256 Number of ways to tile an n X n square using rectangles with distinct height X width dimensions.

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A360498 Number of ways to tile an n X n square using oblongs with distinct dimensions.

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Examples

A334088 a(n) = sqrt(Resultant(T(2n,x/2), T(2n,i*x/2))), where T(n,x) is a Chebyshev polynomial of the first kind and i = sqrt(-1).

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).