A332352 -id:A332352 - cp's OEIS frontend

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.

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
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See A332367.
```

A332353 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

Original entry on oeis.org

0, 0, 0, 1, 2, 8, 2, 4, 14, 24, 3, 6, 22, 38, 60, 4, 8, 30, 52, 82, 112, 5, 10, 40, 70, 112, 154, 212, 6, 12, 50, 88, 142, 196, 270, 344, 7, 14, 62, 110, 178, 246, 340, 434, 548, 8, 16, 74, 132, 214, 296, 410, 524, 662, 800, 9, 18, 88, 158, 258, 358, 498, 638, 808, 978, 1196

Offset: 1

N. J. A. Sloane, Feb 10 2020

This is the triangle in A332352, halved.

			Triangle begins:
0,
0, 0,
1, 2, 8,
2, 4, 14, 24,
3, 6, 22, 38, 60,
4, 8, 30, 52, 82, 112,
5, 10, 40, 70, 112, 154, 212,
6, 12, 50, 88, 142, 196, 270, 344,
7, 14, 62, 110, 178, 246, 340, 434, 548,
8, 16, 74, 132, 214, 296, 410, 524, 662, 800,
...

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)/2.

The main diagonal is A177719.

Cf. A332350, A332352.

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)/2,n=1..m),); od:

A332353[m_,n_]:=Sum[If[GCD[i,j]==2,2(m-i)(n-j),0],{i,2,m-1,2},{j,2,n-1,2}]+If[n>2,m*n-2m,0]+If[m>2,m*n-2n,0];Table[A332353[m, n],{m,15},{n, m}] (* Paolo Xausa, Oct 18 2023 *)

A332363 Triangle read by rows: T(m,n) = number of unstable threshold functions (the function u_{0,1}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

Original entry on oeis.org

1, 2, 7, 3, 11, 19, 4, 18, 31, 51, 5, 24, 42, 69, 95, 6, 33, 59, 98, 135, 191, 7, 41, 74, 124, 172, 243, 311, 8, 52, 94, 158, 219, 310, 397, 507, 9, 62, 114, 191, 265, 376, 482, 615, 747, 10, 75, 138, 233, 325, 462, 593, 758, 921, 1135

Offset: 2

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
1,
2, 7,
3, 11, 19,
4, 18, 31, 51,
5, 24, 42, 69, 95,
6, 33, 59, 98, 135, 191,
7, 41, 74, 124, 172, 243, 311,
8, 52, 94, 158, 219, 310, 397, 507,
9, 62, 114, 191, 265, 376, 482, 615, 747,
10, 75, 138, 233, 325, 462, 593, 758, 921, 1135,
...

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 11.

Cf. A332350, A332352, A331781.

Main diagonal is A332364.

VQ := proc(m,n,q) local eps,a,i,j; eps := 10^(-6); a:=0;
for i from ceil(-m+eps) to floor(m-eps) do
for j from ceil(-n+eps) to floor(n-eps) 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;
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; # A331781
u01:=(m,n) -> 2*VQ(m/2,n/2,1)+2-VS(m,n); # This sequence
for m from 2 to 12 do lprint([seq(u01(m,n),n=2..m)]); od:

A332365 Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

Original entry on oeis.org

3, 6, 13, 9, 21, 33, 12, 30, 49, 73, 15, 40, 66, 99, 133, 18, 51, 85, 130, 177, 237, 21, 63, 106, 164, 224, 301, 381, 24, 76, 130, 202, 277, 374, 475, 593, 27, 90, 154, 241, 331, 448, 570, 713, 857, 30, 105, 182, 287, 395, 538, 687, 862, 1039, 1261, 33, 121, 211, 335, 462, 632, 808, 1016, 1226, 1489, 1757

Offset: 2

N. J. A. Sloane, Feb 11 2020

			Triangle begins:
3,
6, 13,
9, 21, 33,
12, 30, 49, 73,
15, 40, 66, 99, 133,
18, 51, 85, 130, 177, 237,
21, 63, 106, 164, 224, 301, 381,
24, 76, 130, 202, 277, 374, 475, 593,
27, 90, 154, 241, 331, 448, 570, 713, 857,
...

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 11.

Cf. A332350, A332352, A332363, A331781.

Main diagonal is A332366.

VQ := proc(m,n,q) local eps,a,i,j; eps := 10^(-6); a:=0;
for i from ceil(-m+eps) to floor(m-eps) do
for j from ceil(-n+eps) to floor(n-eps) 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;
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; # A331781
u02:=(m,n) -> VQ(m,n,2)+2-2*VQ(m/2,n/2,1)+VS(m,n); # This sequence
for m from 2 to 12 do lprint([seq(u02(m,n),n=2..m)]); od:

cp's OEIS Frontend

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.

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A332353 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

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A332363 Triangle read by rows: T(m,n) = number of unstable threshold functions (the function u_{0,1}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

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Maple

A332365 Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

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cp's OEIS Frontend

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

A332353 Triangle read by rows: T(m,n) = Sum_{-m= n >= 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A332363 Triangle read by rows: T(m,n) = number of unstable threshold functions (the function u_{0,1}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

A332365 Triangle read by rows: T(m,n) = number of threshold functions (the function u_{0,2}(m,n) of Alekseyev et al. 2015) for m >= n >= 2.

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Author

Keywords

Examples

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Maple

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.