A332356 -id:A332356 - cp's OEIS frontend

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.

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).

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:

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.

A332361 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 vertices in the partition, for m >= n >= 1.

Original entry on oeis.org

3, 4, 6, 5, 9, 14, 6, 13, 22, 36, 7, 18, 31, 52, 76, 8, 24, 43, 74, 110, 160, 9, 31, 56, 97, 144, 210, 276, 10, 39, 72, 126, 188, 275, 363, 478, 11, 48, 89, 157, 235, 345, 456, 601, 756, 12, 58, 109, 193, 290, 427, 565, 745, 938, 1164, 13, 69, 130, 231, 347, 511, 675, 890, 1120, 1390, 1660

Offset: 1

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
3,
4, 6,
5, 9, 14,
6, 13, 22, 36,
7, 18, 31, 52, 76,
8, 24, 43, 74, 110, 160,
9, 31, 56, 97, 144, 210, 276,
10, 39, 72, 126, 188, 275, 363, 478,
11, 48, 89, 157, 235, 345, 456, 601, 756,
12, 58, 109, 193, 290, 427, 565, 745, 938, 1164,
...

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-A331360.

Main diagonal is A332362.

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; # A332354
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; # A332356
ct := (m,n) -> ct3(m,n) + ct4(m,n); # A332357
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; # A332359
ctv := (m,n) -> cte(m,n) - ct(m,n) + 1; # A332361
for m from 1 to 12 do lprint([seq(ctv(m,n),n=1..m)]); od:

T(m,n) = A332359(m,n) - A332357(m,n) + 1 (Euler's formula).

cp's OEIS Frontend

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.

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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.

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Maple

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.

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Maple

Formula

A332361 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 vertices in the partition, for m >= n >= 1.

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Author

Keywords

Examples

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Programs

Maple

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cp's OEIS Frontend

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.

Views

Author

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Examples

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Programs

Maple

Formula

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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Author

Keywords

Examples

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Programs

Maple

A332359 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 edges in the partition, for m >= n >= 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A332361 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 vertices in the partition, for m >= n >= 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

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.

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.

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.

A332361 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 vertices in the partition, for m >= n >= 1.