A332350 -id:A332350 - cp's OEIS frontend

A332352 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

0, 0, 0, 2, 4, 16, 4, 8, 28, 48, 6, 12, 44, 76, 120, 8, 16, 60, 104, 164, 224, 10, 20, 80, 140, 224, 308, 424, 12, 24, 100, 176, 284, 392, 540, 688, 14, 28, 124, 220, 356, 492, 680, 868, 1096, 16, 32, 148, 264, 428, 592, 820, 1048, 1324, 1600, 18, 36, 176, 316, 516, 716, 996, 1276, 1616, 1956, 2392

Offset: 1

N. J. A. Sloane, Feb 10 2020

			Triangle begins:
0,
0, 0,
2, 4, 16,
4, 8, 28, 48,
6, 12, 44, 76, 120,
8, 16, 60, 104, 164, 224,
10, 20, 80, 140, 224, 308, 424,
12, 24, 100, 176, 284, 392, 540, 688,
14, 28, 124, 220, 356, 492, 680, 868, 1096,
16, 32, 148, 264, 428, 592, 820, 1048, 1324, 1600,
...

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
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. This sequence is f_2(m,n).

The main diagonal is A331772.

Cf. A332350, A332351, A332353.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
for m from 1 to 12 do lprint(seq(VR(m,n,2),n=1..m),); od:

A332352[m_,n_]:=Sum[If[GCD[i,j]==2,4(m-i)(n-j),0],{i,2,m-1,2},{j,2,n-1,2}]+If[n>2,2(m*n-2m),0]+If[m>2,2(m*n-2n),0];Table[A332352[m, n],{m,15},{n, m}] (* Paolo Xausa, Oct 18 2023 *)

A332357 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of cells (both 3-sided and 4-sided) in the partition, for m >= n >= 1.

Original entry on oeis.org

1, 2, 5, 3, 9, 17, 4, 14, 28, 47, 5, 20, 41, 70, 105, 6, 27, 57, 99, 150, 215, 7, 35, 75, 131, 199, 286, 381, 8, 44, 96, 169, 258, 372, 497, 649, 9, 54, 119, 211, 323, 467, 625, 817, 1029, 10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563, 11, 77, 173, 309, 475, 689, 923, 1208, 1523, 1878, 2257

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
1,
2, 5,
3, 9, 17,
4, 14, 28, 47,
5, 20, 41, 70, 105,
6, 27, 57, 99, 150, 215,
7, 35, 75, 131, 199, 286, 381,
8, 44, 96, 169, 258, 372, 497, 649,
9, 54, 119, 211, 323, 467, 625, 817, 1029,
10, 65, 145, 258, 396, 574, 769, 1006, 1268, 1563,
...

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. See Theorem 13.
N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Cf. A332350, A332352, A332354, A332359 (edges).

Main diagonal is A332358.

Maple
```
See A332354 and A332356.
```

T(m,n) = A332354(m,n)+A332356(m,n).

A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

Original entry on oeis.org

4, 8, 20, 12, 32, 52, 16, 48, 80, 124, 20, 64, 108, 168, 228, 24, 84, 144, 228, 312, 428, 28, 104, 180, 288, 396, 544, 692, 32, 128, 224, 360, 496, 684, 872, 1100, 36, 152, 268, 432, 596, 824, 1052, 1328, 1604, 40, 180, 320, 520, 720, 1000, 1280, 1620, 1960, 2396

Offset: 2

N. J. A. Sloane, Feb 12 2020

			Triangle begins:
4,
8, 20,
12, 32, 52,
16, 48, 80, 124,
20, 64, 108, 168, 228,
24, 84, 144, 228, 312, 428,
28, 104, 180, 288, 396, 544, 692,
32, 128, 224, 360, 496, 684, 872, 1100,
36, 152, 268, 432, 596, 824, 1052, 1328, 1604,
...

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. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332371, A332372, A332374.

For main diagonal see A332368.

# Maple code for sequences mentioned in Theorem 12 of Alekseyev et al. (2015).
VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
VS := proc(m,n) local a,i,j; a:=0; # A331781
for i from 1 to m-1 do for j from 1 to n-1 do
if gcd(i,j)=1 then a:=a+1; fi; od: od: a; end;
c3 := (m,n) -> VR(m,n,2)+4; # A332367
for m from 2 to 12 do lprint([seq(c3(m,n),n=2..m)]); od:
[seq(c3(n,n)/4,n=2..40)]; # A332368
c4 := (m,n) -> VR(m,n,1)/2 - VR(m,n,2) - 3; # A332369
for m from 2 to 12 do lprint([seq(c4(m,n),n=2..m)]); od:
[seq(c4(n,n),n=2..40)]; # A332370
ct := (m,n) -> c3(m,n)+c4(m,n); # A332371
for m from 2 to 12 do lprint([seq(ct(m,n),n=2..m)]); od:
[seq(ct(n,n),n=2..40)]; # A114043
et := (m,n) -> VR(m,n,1) - VR(m,n,2)/2 - VS(m,n) - 2; # A332372
for m from 2 to 12 do lprint([seq(et(m,n),n=2..m)]); od:
[seq(et(n,n),n=2..40)]; # A332373
vt := (m,n) ->  et(m,n) - ct(m,n) +1; # A332374
for m from 2 to 12 do lprint([seq(vt(m,n),n=2..m)]); od:
[seq(vt(n,n),n=2..40)]; # A332375

A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

Original entry on oeis.org

7, 14, 29, 23, 50, 87, 34, 75, 132, 201, 47, 106, 189, 290, 419, 62, 141, 252, 387, 560, 749, 79, 182, 327, 504, 731, 980, 1283, 98, 227, 410, 633, 920, 1235, 1618, 2041, 119, 278, 503, 778, 1133, 1522, 1995, 2518, 3107, 142, 333, 604, 935, 1362, 1829, 2398, 3027, 3736, 4493

Offset: 2

N. J. A. Sloane, Feb 12 2020

Equals sum of triangles A332367 and A332369.

			Triangle begins:
7,
14, 29,
23, 50, 87,
34, 75, 132, 201,
47, 106, 189, 290, 419,
62, 141, 252, 387, 560, 749,
79, 182, 327, 504, 731, 980, 1283,
98, 227, 410, 633, 920, 1235, 1618, 2041,
119, 278, 503, 778, 1133, 1522, 1995, 2518, 3107,
...

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. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332367, A332372, A332374.

For main diagonal see A114043.

Maple
```
See A332367.
```

A332354 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of triangular cells in the partition, for m >= n >= 1.

Original entry on oeis.org

1, 2, 5, 3, 8, 15, 4, 11, 22, 33, 5, 14, 31, 48, 71, 6, 17, 40, 63, 94, 125, 7, 20, 51, 82, 125, 168, 227, 8, 23, 62, 101, 156, 211, 286, 361, 9, 26, 75, 124, 193, 262, 357, 452, 567, 10, 29, 88, 147, 230, 313, 428, 543, 682, 821, 11, 32, 103, 174, 275, 376, 517, 658, 829, 1000, 1219

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
1,
2, 5,
3, 8, 15,
4, 11, 22, 33,
5, 14, 31, 48, 71,
6, 17, 40, 63, 94, 125,
7, 20, 51, 82, 125, 168, 227,
8, 23, 62, 101, 156, 211, 286, 361,
9, 26, 75, 124, 193, 262, 357, 452, 567,
10, 29, 88, 147, 230, 313, 428, 543, 682, 821,
...

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. See Theorem 13.
N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Cf. A332350, A332352, A332356.

Main diagonal is A332355.

# VR(m,n,q) is f_q(m,n) from the Alekseyev et al. reference.
VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
ct3 := proc(m,n) local i; global VR;
if m=1 or n=1 then max(m,n) else VR(m,n,2)/2+m+n+1; fi; end;
for m from 1 to 12 do lprint([seq(ct3(m,n),n=1..m)]); od:

A332372 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of edges in the partition for m >= n >= 2.

Original entry on oeis.org

9, 20, 43, 35, 77, 139, 54, 118, 213, 327, 77, 170, 310, 479, 703, 104, 229, 417, 642, 941, 1259, 135, 299, 546, 842, 1236, 1657, 2183, 170, 376, 688, 1062, 1561, 2094, 2759, 3487, 209, 464, 850, 1313, 1933, 2594, 3418, 4321, 5355, 252, 559, 1024, 1581, 2327, 3118, 4107, 5190, 6431, 7723

Offset: 2

N. J. A. Sloane, Feb 12 2020

			Triangle begins:
9,
20, 43,
35, 77, 139,
54, 118, 213, 327,
77, 170, 310, 479, 703,
104, 229, 417, 642, 941, 1259,
135, 299, 546, 842, 1236, 1657, 2183,
170, 376, 688, 1062, 1561, 2094, 2759, 3487,
209, 464, 850, 1313, 1933, 2594, 3418, 4321, 5355,
...

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. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332367, A332371, A332374.

For main diagonal see A332373.

Maple
```
See A332367.
```

A332374 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of vertices in the partition for m >= n >= 2.

Original entry on oeis.org

3, 7, 15, 13, 28, 53, 21, 44, 82, 127, 31, 65, 122, 190, 285, 43, 89, 166, 256, 382, 511, 57, 118, 220, 339, 506, 678, 901, 73, 150, 279, 430, 642, 860, 1142, 1447, 91, 187, 348, 536, 801, 1073, 1424, 1804, 2249, 111, 227, 421, 647, 966, 1290, 1710, 2164, 2696, 3231

Offset: 2

N. J. A. Sloane, Feb 12 2020

T(m,n) = A332372(m,n) - A332371(m,n) + 1 (this is Euler's formula).

			Triangle begins:
3,
7, 15,
13, 28, 53,
21, 44, 82, 127,
31, 65, 122, 190, 285,
43, 89, 166, 256, 382, 511,
57, 118, 220, 339, 506, 678, 901,
73, 150, 279, 430, 642, 860, 1142, 1447,
91, 187, 348, 536, 801, 1073, 1424, 1804, 2249,
...

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. See Theorem 12.
N. J. A. Sloane, Illustration for m=n=3

Cf. A332350, A332352, A331781, A332367, A332371, A332372.

For main diagonal see A332375.

Maple
```
See A332367.
```

A332351 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

Original entry on oeis.org

0, 1, 6, 2, 13, 28, 3, 22, 49, 86, 4, 33, 74, 131, 200, 5, 46, 105, 188, 289, 418, 6, 61, 140, 251, 386, 559, 748, 7, 78, 181, 326, 503, 730, 979, 1282, 8, 97, 226, 409, 632, 919, 1234, 1617, 2040, 9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106, 10, 141, 332, 603, 934, 1361, 1828, 2397, 3026, 3735, 4492

Offset: 1

N. J. A. Sloane, Feb 10 2020

This is the triangle in A332350, halved.

This triangle is the lower half of the array defined in A115009.

			Triangle begins:
0,
1, 6,
2, 13, 28,
3, 22, 49, 86,
4, 33, 74, 131, 200,
5, 46, 105, 188, 289, 418,
6, 61, 140, 251, 386, 559, 748,
7, 78, 181, 326, 503, 730, 979, 1282,
8, 97, 226, 409, 632, 919, 1234, 1617, 2040,
9, 118, 277, 502, 777, 1132, 1521, 1994, 2517, 3106,
...

Jovisa Zunic, Note on the number of two-dimensional threshold functions, SIAM J. Discrete Math. Vol. 25 (2011), No. 3, pp. 1266-1268. See Equation (1.2).

Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)
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. This sequence is f_1(m,n)/2.

The main diagonal is A141255, or A114043 - 1.
Cf. A115009, A332350, A332352.
This is the lower triangle of the array in A115009.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
for m from 1 to 12 do lprint(seq(VR(m,n,1)/2,n=1..m),); od:

A332351[m_,n_]:=Sum[If[CoprimeQ[i,j],2(m-i)(n-j),0],{i,m-1},{j,n-1}]+2m*n-m-n;Table[A332351[m,n],{m,15},{n,m}] (* Paolo Xausa, Oct 18 2023 *)

A332356 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of quadrilateral cells in the partition, for m >= n >= 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 0, 3, 6, 14, 0, 6, 10, 22, 34, 0, 10, 17, 36, 56, 90, 0, 15, 24, 49, 74, 118, 154, 0, 21, 34, 68, 102, 161, 211, 288, 0, 28, 44, 87, 130, 205, 268, 365, 462, 0, 36, 57, 111, 166, 261, 341, 463, 586, 742, 0, 45, 70, 135, 200, 313, 406, 550, 694, 878, 1038

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
0,
0, 0,
0, 1, 2,
0, 3, 6, 14,
0, 6, 10, 22, 34,
0, 10, 17, 36, 56, 90,
0, 15, 24, 49, 74, 118, 154,
0, 21, 34, 68, 102, 161, 211, 288,
0, 28, 44, 87, 130, 205, 268, 365, 462,
0, 36, 57, 111, 166, 261, 341, 463, 586, 742,
...

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. See Theorem 13.
N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Cf. A332350, A332352, A332354.

Main diagonal is A324043.

# VR(m,n,q) is f_q(m,n) from the Alekseyev et al. reference.
VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
ct4 := proc(m,n) local i; global VR;
if m=1 or n=1 then 0 else VR(m,n,1)/4-VR(m,n,2)/2-m/2-n/2-1; fi; end;
for m from 1 to 12 do lprint([seq(ct4(m,n),n=1..m)]); od:

VR[m_, n_, q_] := Module[{a = 0, i, j}, For[i = -m + 1, i <= m - 1, i++, For[j = -n + 1, j <= n - 1, j++, If[GCD[i, j] == q, a = a + (m - Abs[i])*(n - Abs[j])]]];a];
ct4 [m_, n_] := If[m == 1 || n == 1, 0, VR[m, n, 1]/4 - VR[m, n, 2]/2 - m/2 - n/2 - 1];
Table[ct4[m, n], {m, 1, 12}, {n, 1, m}] // Flatten (* Jean-François Alcover, Mar 09 2023, after Maple code *)

A332359 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of edges in the partition, for m >= n >= 1.

Original entry on oeis.org

3, 5, 10, 7, 17, 30, 9, 26, 49, 82, 11, 37, 71, 121, 180, 13, 50, 99, 172, 259, 374, 15, 65, 130, 227, 342, 495, 656, 17, 82, 167, 294, 445, 646, 859, 1126, 19, 101, 207, 367, 557, 811, 1080, 1417, 1784, 21, 122, 253, 450, 685, 1000, 1333, 1750, 2205, 2726, 23, 145, 302, 539, 821, 1199, 1597, 2097, 2642, 3267, 3916

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
3,
5, 10,
7, 17, 30,
9, 26, 49, 82,
11, 37, 71, 121, 180,
13, 50, 99, 172, 259, 374,
15, 65, 130, 227, 342, 495, 656,
17, 82, 167, 294, 445, 646, 859, 1126,
19, 101, 207, 367, 557, 811, 1080, 1417, 1784,
21, 122, 253, 450, 685, 1000, 1333, 1750, 2205, 2726,
...

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. See Theorem 13.
N. J. A. Sloane, Illustration for (m,n) = (2,2), (3,1), (3,2), (3,3) [c_3 = number of triangles, c_4 = number of quadrilaterals; c, e, v = numbers of cells, edges, vertices]

Cf. A332350, A332352, A332354, A332357 (edges).

Main diagonal is A332360.

VR := proc(m,n,q) local a,i,j; a:=0;
for i from -m+1 to m-1 do for j from -n+1 to n-1 do
if gcd(i,j)=q then a:=a+(m-abs(i))*(n-abs(j)); fi; od: od: a; end;
cte := proc(m,n) local i; global VR;
if m=1 or n=1 then 2*max(m,n)+1 else VR(m,n,1)/2-VR(m,n,2)/4+m+n; fi; end;
for m from 1 to 12 do lprint([seq(cte(m,n),n=1..m)]); od:

T(m,n) = (3*A332354(m,n) + 4*A332356(m,n) + m + n + 1)/2.

cp's OEIS Frontend

A332352 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

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A332357 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of cells (both 3-sided and 4-sided) in the partition, for m >= n >= 1.

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A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

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Maple

A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

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A332354 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1*x_1 + a_2*x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of triangular cells in the partition, for m >= n >= 1.

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Maple

A332372 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of edges in the partition for m >= n >= 2.

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Maple

A332374 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1*x_1 + a_2*x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of vertices in the partition for m >= n >= 2.

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Maple

A332351 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

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Mathematica

A332357 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of cells (both 3-sided and 4-sided) in the partition, for m >= n >= 1.

A332367 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = number of triangular cells in the partition for m >= n >= 2.

A332371 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of cells in the partition for m >= n >= 2.

A332354 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of triangular cells in the partition, for m >= n >= 1.

A332372 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of edges in the partition for m >= n >= 2.

A332374 Consider a partition of the plane (a_1,a_2) in R X R by the lines a_1x_1 + a_2x_2 = 1 for 0 <= x_1 <= m-1, 1 <= x_2 <= 1-1. The cells are (generalized) triangles and quadrilaterals. Triangle read by rows: T(m,n) = total number of vertices in the partition for m >= n >= 2.

A332356 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of quadrilateral cells in the partition, for m >= n >= 1.

A332359 Consider a partition of the triangle with vertices (0, 0), (1, 0), (0, 1) by the lines a_1x_1 + a_2x_2 = 1, where (x_1, x_2) is in {1, 2,...,m} X {1, 2,...,n}, m >= 1, n >= 1. Triangle read by rows: T(m,n) = number of edges in the partition, for m >= n >= 1.