A324042 -id:A324042 - cp's OEIS frontend

A306302 Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).

0, 4, 16, 46, 104, 214, 380, 648, 1028, 1562, 2256, 3208, 4384, 5924, 7792, 10052, 12744, 16060, 19880, 24486, 29748, 35798, 42648, 50648, 59544, 69700, 80992, 93654, 107596, 123374, 140488, 159704, 180696, 203684, 228624, 255892, 285152, 317400, 352096, 389576

Offset: 0

Paarth Jain, Feb 05 2019

nonn

Assuming that the rectangles have vertices at (k,0) and (k,1), k=0..n, the projective map (x,y) -> ((1-y)/(x+1),y/(x+1)) maps their partition to the partition of the right isosceles triangle described by Alekseyev et al. (2015), for which Theorem 13 gives the number of regions, line segments, and intersection points. - Max Alekseyev, Apr 10 2019

The figure is made up of A324042 triangles and A324043 quadrilaterals. - N. J. A. Sloane, Mar 03 2020

Jinyuan Wang, Table of n, a(n) for n = 0..1000
Max Alekseyev, Illustration for n = 3.
M. A. Alekseyev. On the number of two-dimensional threshold functions, arXiv:math/0602511 [math.CO], 2006-2010; doi:10.1137/090750184, SIAM J. Disc. Math. 24(4), 2010, pp. 1617-1631.
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh. On the minimal teaching sets of two-dimensional threshold functions, SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165. doi:10.1137/140978090.
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.
M. Griffiths, Counting the regions in a regular drawing of K_{n,n}, J. Int. Seq. 13 (2010) # 10.8.5, Lemma 2.
Robert Israel, Maple program, Feb 07 2019
S. Legendre, The Number of Crossings in a Regular Drawing of the Complete Bipartite Graph, J. Integer Seqs., Vol. 12, 2009.
Scott R. Shannon, Colored illustration for T(1,1)
Scott R. Shannon, Colored illustration for T(2,1)
Scott R. Shannon, Colored illustration for T(3,1)
Scott R. Shannon, Colored illustration for T(4,1)
Scott R. Shannon, Colored illustration for T(5,1)
Scott R. Shannon, Colored illustration for T(6,1)
Scott R. Shannon, Colored illustration for T(7,1)
Scott R. Shannon, Colored illustration for T(8,1)
Scott R. Shannon, Colored illustration for T(9,1)
Scott R. Shannon, Colored illustration for T(10,1)
Scott R. Shannon, Colored illustration for T(11,1)
Scott R. Shannon, Colored illustration for T(12,1)
Scott R. Shannon, Colored illustration for T(13,1)
Scott R. Shannon, Colored illustration for T(14,1)
Scott R. Shannon, Colored illustration for T(15,1)
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
Index entries for sequences related to stained glass windows

See A331755 for the number of vertices, A331757 for the number of edges.
Cf. A007678, A108914, A114043, A324042, A324043, A333286, A333287, A333288, A334694.
A column of A288187. See A288177 for additional references.
Also a column of A331452 and A356790.
The following eight sequences are all essentially the same. The simplest is A115004(n), which we denote by z(n). Then A088658(n) = 4*z(n-1); A114043(n) = 2*z(n-1)+2*n^2-2*n+1; A114146(n) = 2*A114043(n); A115005(n) = z(n-1)+n*(n-1); A141255(n) = 2*z(n-1)+2*n*(n-1); A290131(n) = z(n-1)+(n-1)^2; A306302(n) = z(n)+n^2+2*n. - N. J. A. Sloane, Feb 04 2020

# Maple from N. J. A. Sloane, Mar 04 2020, starting at n=1:  First define z(n) = A115004
z := proc(n)
    local a, b, r ;
    r := 0 ;
    for a from 1 to n do
    for b from 1 to n do
        if igcd(a, b) = 1 then
            r := r+(n+1-a)*(n+1-b);
        end if;
    end do:
    end do:
    r ;
end proc:
a := n-> z(n)+n^2+2*n;
[seq(a(n), n=1..50)];

z[n_] := Sum[(n - i + 1)(n - j + 1) Boole[GCD[i, j] == 1], {i, n}, {j, n}];
a[0] = 0;
a[n_] := z[n] + n^2 + 2n;
a /@ Range[0, 40] (* Jean-François Alcover, Mar 24 2020 *)

from sympy import totient
def A306302(n): return 2*n*(n+1) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(2,n+1)) # Chai Wah Wu, Aug 16 2021

a(n) = n + (A114043(n+1) - 1)/2, conjectured by N. J. A. Sloane, Feb 07 2019; proved by Max Alekseyev, Apr 10 2019
a(n) = n + A115005(n+1) = n + A141255(n+1)/2. - Max Alekseyev, Apr 10 2019
a(n) = A324042(n) + A324043(n). - Jinyuan Wang, Mar 19 2020
a(n) = Sum_{i=1..n, j=1..n, gcd(i,j)=1} (n+1-i)*(n+1-j) + n^2 + 2*n. - N. J. A. Sloane, Apr 11 2020
a(n) = 2n(n+1) + Sum_{i=2..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

a(6)-a(20) from Robert Israel, Feb 07 2019
Edited and more terms added by Max Alekseyev, Apr 10 2019
a(0) added by N. J. A. Sloane, Feb 04 2020

A331757 Number of edges in a figure made up of a row of n adjacent congruent rectangles upon drawing diagonals of all possible rectangles.

Original entry on oeis.org

8, 28, 80, 178, 372, 654, 1124, 1782, 2724, 3914, 5580, 7626, 10352, 13590, 17540, 22210, 28040, 34670, 42760, 51962, 62612, 74494, 88508, 104042, 121912, 141534, 163664, 187942, 215636, 245490, 279260, 316022, 356456, 399898, 447612, 498698, 555352

Offset: 1

N. J. A. Sloane, Feb 04 2020

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2020). Also arXiv:2009.07918.

A306302 gives number of regions in the figure.
This is column 1 of A331454.
Cf. A324042, A324043.

Table[n^2 + 4n + 1 + Sum[Sum[(2 * Boole[GCD[i, j] == 1] - Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}], {n, 1, 37}] (* Joshua Oliver, Feb 05 2020 *)

from sympy import totient
def A331757(n): return 8 if n == 1 else 2*(n*(n+3) + sum(totient(i)*(n+1-i)*(n+1+i) for i in range(2,n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1,n+1))) # Chai Wah Wu, Aug 16 2021

a(n) = (2*n + 2 + 3*A324042(n) + 4*A324043(n))/2 [Corrected by Chai Wah Wu, Aug 16 2021]

For n > 1, a(n) = 2*(n*(n+3) + Sum_{i=2..floor(n/2)} (n+1-i)*(n+1+i)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2*n+2-i)*phi(i)). - Chai Wah Wu, Aug 16 2021

A324043 Number of quadrilateral regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.

Original entry on oeis.org

0, 2, 14, 34, 90, 154, 288, 462, 742, 1038, 1512, 2074, 2904, 3774, 4892, 6154, 7864, 9662, 12022, 14638, 17786, 20998, 25024, 29402, 34672, 40038, 46310, 53038, 61090, 69454, 79344, 89890, 101792, 113854, 127476, 141866, 158428, 175182, 193760, 213274, 235444, 258182, 283858, 310750, 339986

Offset: 1

Jinyuan Wang, May 01 2019

nonn

A row of n adjacent congruent rectangles can only be divided into triangles (cf. A324042) or quadrilaterals when drawing diagonals. Proof is given in Alekseyev et al. (2015) under the transformation described in A306302.

			For k adjacent congruent rectangles, the number of quadrilateral regions in the j-th rectangle is:
k\j|  1   2   3   4   5   6   7  ...
---+--------------------------------
1  |  0,  0,  0,  0,  0,  0,  0, ...
2  |  1,  1,  0,  0,  0,  0,  0, ...
3  |  3,  8,  3,  0,  0,  0,  0, ...
4  |  5, 12, 12,  5,  0,  0,  0, ...
5  |  7, 22, 32, 22,  7,  0,  0, ...
6  |  9, 28, 40, 40, 28,  9,  0, ...
7  | 11, 38, 58, 74, 58, 38, 11, ...
...
a(4) = 5 + 12 + 12 + 5 = 34.

Chai Wah Wu, Table of n, a(n) for n = 1..10000
M. A. Alekseyev, M. Basova, and N. Yu. Zolotykh, On the minimal teaching sets of two-dimensional threshold functions. SIAM Journal on Discrete Mathematics 29:1 (2015), 157-165.
Lars Blomberg, Scott R. Shannon, and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Robert Israel, Maple program.
Jinyuan Wang, Illustration for n = 1, 2, 3, 4, 5.

Cf. A007678, A108914, A114043, A115005, A177719, A306302, A324042, A333286, A333287, A333288.

See Robert Israel link.
There are also Maple programs for both A306302 and A324042. Then a := n -> A306302(n) - A324042(n); # N. J. A. Sloane, Mar 04 2020

Table[Sum[Sum[(Boole[GCD[i, j] == 1] - 2 * Boole[GCD[i, j] == 2]) * (n + 1 - i) * (n + 1 - j), {j, 1, n}], {i, 1, n}] - n^2, {n, 1, 45}] (* Joshua Oliver, Feb 05 2020 *)

{ A324043(n) = sum(i=1, n, sum(j=1, n, ( (gcd(i, j)==1) - 2*(gcd(i,j)==2) ) * (n+1-i) * (n+1-j) )) - n^2; } \\ Max Alekseyev, Jul 08 2019

from sympy import totient
def A324043(n): return 0 if n==1 else -2*(n-1)**2 + sum(totient(i)*(n+1-i)*(7*i-2*n-2) for i in range(2,n//2+1)) + sum(totient(i)*(n+1-i)*(2*n+2-i) for i in range(n//2+1,n+1)) # Chai Wah Wu, Aug 16 2021

a(n) = A115005(n+1) - A177719(n+1) - n - 1 = Sum_{i,j=1..n; gcd(i,j)=1} (n+1-i)*(n+1-j) - 2*Sum_{i,j=1..n; gcd(i,j)=2} (n+1-i)*(n+1-j) - n^2. - Max Alekseyev, Jul 08 2019
a(n) = A306302(n) - A324042(n).
For n>1, a(n) = -2(n-1)^2 + Sum_{i=2..floor(n/2)} (n+1-i)*(7i-2n-2)*phi(i) + Sum_{i=floor(n/2)+1..n} (n+1-i)*(2n+2-i)*phi(i). - Chai Wah Wu, Aug 16 2021

a(8)-a(23) from Robert Israel, Jul 07 2019

Terms a(24) onward from Max Alekseyev, Jul 08 2019

A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

Original entry on oeis.org

4, 8, 8, 12, 22, 12, 16, 36, 36, 16, 20, 52, 70, 52, 20, 24, 66, 100, 100, 66, 24, 28, 82, 134, 160, 134, 82, 28, 32, 98, 166, 218, 218, 166, 98, 32, 36, 116, 198, 276, 310, 276, 198, 116, 36, 40, 134, 230, 328, 396, 396, 328, 230, 134, 40, 44, 154, 266, 386

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.
Since the cells are either triangles or quadrilaterals, this is the sum of the two arrays A333286 and A333287.
It would be nice to have a formula for these entries. It is easy to see that the first column is 4n for n>=1.

			Triangle begins:
   4;
   8,   8;
  12,  22,  12;
  16,  36,  36,  16;
  20,  52,  70,  52,  20;
  24,  66, 100, 100,  66,  24;
  28,  82, 134, 160, 134,  82,  28;
  ...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333286, A333287, A333288, A335056, A335074.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A333286 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of triangular regions in the k-th rectangle.

Original entry on oeis.org

4, 7, 7, 9, 14, 9, 11, 24, 24, 11, 13, 30, 38, 30, 13, 15, 38, 60, 60, 38, 15, 17, 44, 76, 86, 76, 44, 17, 19, 52, 92, 120, 120, 92, 52, 19, 21, 58, 106, 146, 158, 146, 106, 58, 21, 23, 66, 126, 178, 216, 216, 178, 126, 66, 23, 25, 72, 142, 206, 264, 278

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.

It would be nice to have a formula for these entries. It is easy to see that the first column is 2n+3 for n>1.

			Triangle begins:
4,
7,  7,
9, 14,  9,
11, 24, 24, 11,
13, 30, 38, 30, 13,
15, 38, 60, 60, 38, 15,
17, 44, 76, 86, 76, 44, 17,
...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333287, A333288.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A333287 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of quadrilateral regions in the k-th rectangle.

Original entry on oeis.org

0, 1, 1, 3, 8, 3, 5, 12, 12, 5, 7, 22, 32, 22, 7, 9, 28, 40, 40, 28, 9, 11, 38, 58, 74, 58, 38, 11, 13, 46, 74, 98, 98, 74, 46, 13, 15, 58, 92, 130, 152, 130, 92, 58, 15, 17, 68, 104, 150, 180, 180, 150, 104, 68, 17, 19, 82, 124, 180, 224, 254, 224, 180, 124, 82, 19

Offset: 1

N. J. A. Sloane, Mar 20 2020

This was originally based on the data in Jinyuan Wang's A324042, and then extended by Lars Blomberg.

It would be nice to have a formula for these entries. It is easy to see that the first column is 2n-3 for n>1.

			Triangle begins:
0,
1,  1,
3,  8,  3,
5, 12, 12,  5,
7, 22, 32, 22,  7,
9, 28, 40, 40, 28,  9,
11, 38, 58, 74, 58, 38, 11,
...

Lars Blomberg, Table of n, a(n) for n = 1..3240 (the first 80 rows)
Lars Blomberg, Scott R. Shannon and N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.

Cf. A306302, A331452, A324042, A324043, A333286, A333288.

a(29) and beyond from Lars Blomberg, Apr 23 2020

A369178 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=3, in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

Original entry on oeis.org

2, 8, 4, 22, 14, 52, 34, 98, 82, 184, 146, 302, 268, 484, 426, 8, 710, 694, 4, 1064, 986, 8, 1498, 1436, 12, 2056, 1986, 12, 2710, 2780, 12, 3624, 3630, 24, 4682, 4728, 20, 6012, 5970, 24, 7518, 7628, 28, 9408, 9406, 32, 11526, 11702, 40, 14028, 14246, 64, 16782, 17330, 60

Offset: 1

Scott R. Shannon, Jan 15 2024

Unlike the graph in A306302, or the complete bipartite graph of order n, for n>=8 the graph contains regions with 5 edges. It is likely 5 is the maximum number of edges in any region for all n.

			The table begins:
2;
8, 4;
22, 14;
52, 34;
98, 82;
184, 146;
302, 268;
484, 426, 8;
710, 694, 4;
1064, 986, 8;
1498, 1436, 12;
2056, 1986, 12;
2710, 2780, 12;
3624, 3630, 24;
4682, 4728, 20;
6012, 5970, 24;
7518, 7628, 28;
9408, 9406, 32;
11526, 11702, 40;
14028, 14246, 64;
16782, 17330, 60;
20220, 20518, 68;
23998, 24468, 80;
28304, 28786, 84;
.
.

Scott R. Shannon, Image for n = 7.
Scott R. Shannon, Image for n = 8.

Cf. A369175 (regions), A369176 (vertices), A369177 (edges), A306302, A324042, A324043, A368758.

Sum of row(n) = A369175(n).

A332610 Triangle read by rows: T(m,n) = number of triangular regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

Original entry on oeis.org

4, 14, 48, 32, 102, 128, 70, 192, 204, 288, 124, 326, 312, 396, 512, 226, 524, 516, 600, 716, 928, 360, 802, 784, 868, 984, 1196, 1472, 566, 1192, 1196, 1280, 1396, 1608, 1884, 2304, 820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328, 1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

See A331457 for illustrations.

			Triangle begins:
[4],
[14, 48],
[32, 102, 128],
[70, 192, 204, 288],
[124, 326, 312, 396, 512],
[226, 524, 516, 600, 716, 928],
[360, 802, 784, 868, 984, 1196, 1472],
[566, 1192, 1196, 1280, 1396, 1608, 1884, 2304],
[820, 1634, 1704, 1788, 1904, 2116, 2392, 2812, 3328],
[1218, 2296, 2500, 2584, 2700, 2912, 3188, 3608, 4124, 4928],
[1696, 3074, 3456, 3540, 3656, 3868, 4144, 4564, 5080, 5884, 6848],
[2310, 4052, 4684, 4768, 4884, 5096, 5372, 5792, 6308, 7112, 8076, 9312],
...

Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.

The first column is A324042, for which there is an explicit formula.
No formula is known for column 2, which is A332606.
For m>=n>=3 we have the (new) theorem that T(m,n) = 4*(m^2+n^2)+12*n+4*m-24 + 4*V(m,m,2)+4*V(n,n,2), where V(m,n,q) = Sum_{i = 1..m, j = 1..n, gcd(i,j)=q} (m+1-i)*(n+1-j).

A332611 Triangle read by rows: T(m,n) = number of quadrilateral regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

Original entry on oeis.org

0, 2, 8, 14, 36, 80, 34, 92, 144, 208, 90, 194, 280, 356, 504, 154, 336, 432, 520, 680, 856, 288, 554, 724, 824, 996, 1184, 1512, 462, 812, 1096, 1208, 1392, 1592, 1932, 2352, 742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640, 1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 12 2020

See A331457 for illustrations.

			Triangle begins:
[0],
[2, 8],
[14, 36, 80],
[34, 92, 144, 208],
[90, 194, 280, 356, 504],
[154, 336, 432, 520, 680, 856],
[288, 554, 724, 824, 996, 1184, 1512],
[462, 812, 1096, 1208, 1392, 1592, 1932, 2352],
[742, 1314, 1680, 1804, 2000, 2212, 2564, 2996, 3640],
[1038, 1756, 2296, 2432, 2640, 2864, 3228, 3672, 4328, 5016],
[1512, 2508, 3268, 3416, 3636, 3872, 4248, 4704, 5372, 6072, 7128],
[2074, 3252, 4416, 4576, 4808, 5056, 5444, 5912, 6592, 7304, 8372, 9616],
....

Cf. A331457, A332599, A332600, A324042, A324043, A332606, A332607, A332595, A332596.

The first column is A324043, for which there is an explicit formula.
No formula is known for column 2, which is A332607.
For m>=n>=3 we have the (new) theorem that T(m,n) = 4*(3*m*n-m-4*n) + 2*(V(m,m,1)-2*V(m,m,2)-m^2-4*m+6) + 2*(V(n,n,1)-2*V(n,n,2)-n^2-4*n+6) where V(m,n,q) = Sum_{i = 1..m, j = 1..n, gcd(i,j)=q} (m+1-i)*(n+1-j).

cp's OEIS Frontend

A306302 Number of regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles (a(0)=0 by convention).

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A331757 Number of edges in a figure made up of a row of n adjacent congruent rectangles upon drawing diagonals of all possible rectangles.

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A324043 Number of quadrilateral regions into which a figure made up of a row of n adjacent congruent rectangles is divided upon drawing diagonals of all possible rectangles.

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A333288 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of regions in the k-th rectangle.

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A333286 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of triangular regions in the k-th rectangle.

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A333287 Triangle read by rows: consider a figure made up of a row of n congruent rectangles and the diagonals of all visible rectangles; T(n,k) (1 <= k <= n) is the number of quadrilateral regions in the k-th rectangle.

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A369178 Irregular table read by rows: T(n,k) is the number of k-sided regions, k>=3, in a graph of n adjacent rectangles in a row with all possible diagonals drawn, as in A306302, but without the rectangles' edges which are perpendicular to the row.

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A332610 Triangle read by rows: T(m,n) = number of triangular regions in a "frame" of size m X n with m >= n >= 1 (see Comments in A331457 for definition of frame).

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