A332354 -id:A332354 - cp's OEIS frontend

A332355 Main diagonal of A332354.

1, 5, 15, 33, 71, 125, 227, 361, 567, 821, 1219, 1697, 2311, 3021, 4019, 5161, 6591, 8197, 10219, 12465, 15111, 18013, 21651, 25625, 30143, 35029, 40955, 47345, 54559, 62285, 71035, 80361, 90807, 101893, 114771, 128417, 143287, 158973, 176915, 195817

Offset: 1

N. J. A. Sloane, Feb 11 2020

nonn

Cf. A332354.

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).

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.

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

A332355 Main diagonal of A332354.

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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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Maple

Formula

A332356 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 quadrilateral cells in the partition, for m >= n >= 1.

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Author

Keywords

Examples

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Maple

Mathematica

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.

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.

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.