A332367 -id:A332367 - cp's OEIS frontend

A332368 Main diagonal of A332367, divided by 4.

1, 5, 13, 31, 57, 107, 173, 275, 401, 599, 837, 1143, 1497, 1995, 2565, 3279, 4081, 5091, 6213, 7535, 8985, 10803, 12789, 15047, 17489, 20451, 23645, 27251, 31113, 35487, 40149, 45371, 50913, 57351, 64173, 71607, 79449, 88419, 97869

Offset: 2

N. J. A. Sloane, Feb 12 2020

nonn

N. J. A. Sloane, Illustration for a(3) = 20/4 = 5

Cf. A332367.

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.
```

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.
```

A332369 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 quadrilateral cells in the partition for m >= n >= 2.

Original entry on oeis.org

3, 6, 9, 11, 18, 35, 18, 27, 52, 77, 27, 42, 81, 122, 191, 38, 57, 108, 159, 248, 321, 51, 78, 147, 216, 335, 436, 591, 66, 99, 186, 273, 424, 551, 746, 941, 83, 126, 235, 346, 537, 698, 943, 1190, 1503, 102, 153, 284, 415, 642, 829, 1118, 1407, 1776, 2097, 123, 186, 345, 504, 777, 1002, 1349, 1696, 2139, 2528, 3047

Offset: 2

N. J. A. Sloane, Feb 12 2020

			Triangle begins:
3,
6, 9,
11, 18, 35,
18, 27, 52, 77,
27, 42, 81, 122, 191,
38, 57, 108, 159, 248, 321,
51, 78, 147, 216, 335, 436, 591,
66, 99, 186, 273, 424, 551, 746, 941,
83, 126, 235, 346, 537, 698, 943, 1190, 1503,...

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, A332374.

For main diagonal see A332370.

Maple
```
See A332367.
```

cp's OEIS Frontend

A332368 Main diagonal of A332367, divided by 4.

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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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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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

A332369 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 quadrilateral cells in the partition for m >= n >= 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

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.

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.

A332369 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 quadrilateral cells in the partition for m >= n >= 2.