A332357 -id:A332357 - cp's OEIS frontend

A332358 Main diagonal of A332357.

1, 5, 17, 47, 105, 215, 381, 649, 1029, 1563, 2257, 3209, 4385, 5925, 7793, 10053, 12745, 16061, 19881, 24487, 29749, 35799, 42649, 50649, 59545, 69701, 80993, 93655, 107597, 123375, 140489, 159705, 180697, 203685, 228625, 255893, 285153, 317401, 352097, 389577

Offset: 1

N. J. A. Sloane, Feb 11 2020

nonn

Lars Blomberg, Scott R. Shannon, N. J. A. Sloane, Graphical Enumeration and Stained Glass Windows, 1: Rectangular Grids, (2021). Also arXiv:2009.07918.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 11.

Cf. A332357, A306302.

a(n) = A306302(n-1) + 1. [see Max Alekseyev's comment in A306302] - Andrey Zabolotskiy, Sep 14 2023

A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

1, 5, 18, 52, 125, 257, 486, 832, 1333, 2027, 3048, 4304, 6057, 8167, 10749, 13929, 18058, 22664, 28533, 34981, 42519, 51425, 62118, 73473, 86768, 101902, 118695, 137138, 159147, 181752, 208813, 237209, 268614, 303718, 340882, 380811, 427540, 477134, 530047

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 04 2020

nonn

The terms are from numeric computation - no formula for a(n) is currently known.
Equivalently, this is also the number of regions formed when all the integer points along the x and y axes with 0 <= x <= n and 0 <= y <= n are joined by straight line segments.
If instead one takes points on the x and y axes with coordinates 1, 1/2, 1/3, 1/4, ..., 1/n, 0, and joins them all by line segments, the resulting figure contains only triangles and quadrilaterals, and the number of regions is given by A332358 (and more generally by A332357 if there are m+1 such points on the x axis and n+1 such points on the y axis).

Lars Blomberg, Table of n, a(n) for n = 1..70
Scott R. Shannon, Illustration for n = 2.
Scott R. Shannon, Illustration for n = 3.
Scott R. Shannon, Illustration for n = 4.
Scott R. Shannon, Illustration for n = 5.
Scott R. Shannon, Illustration for n = 6.
Scott R. Shannon, Illustration for n = 8.
Scott R. Shannon, Illustration for n = 10.
Scott R. Shannon, Illustration for n = 12.
Scott R. Shannon, Illustration for n = 15.
Scott R. Shannon, Illustration for n = 5 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 10 with random distance-based coloring.
Scott R. Shannon, Illustration for n = 15 with random distance-based coloring.

Cf. A333025 (n-gons), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

Original entry on oeis.org

1, 5, 14, 3, 1, 29, 19, 4, 50, 66, 9, 81, 164, 12, 134, 313, 37, 2, 219, 546, 60, 7, 359, 853, 112, 9, 556, 1294, 160, 16, 1, 779, 1940, 283, 43, 3, 1105, 2780, 360, 53, 6, 1540, 3750, 670, 91, 5, 1, 2087, 5064, 873, 132, 11, 2806, 6625, 1144, 164, 7, 3

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

See the links in A332953 for images of the triangles.

			Table begins:
1;
5;
14, 3, 1;
29, 19, 4;
50, 66, 9;
81, 164, 12;
134, 313, 37, 2;
219, 546, 60, 7;
359, 853, 112, 9;
556, 1294, 160, 16, 1;
779, 1940, 283, 43, 3;
1105, 2780, 360, 53, 6;
1540, 3750, 670, 91, 5, 1;
2087, 5064, 873, 132, 11;
2806, 6625, 1144, 164, 7, 3;
The row sums are A332953.

Lars Blomberg, Table of n, a(n) for n = 1..613 (the first 70 rows)

Cf. A332953 (regions), A333026 (vertices), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

3, 10, 33, 96, 235, 486, 933, 1600, 2561, 3884, 5907, 8310, 11793, 15890, 20863, 27002, 35229, 44117, 55820, 68312, 82931, 100368, 121711, 143685, 169750, 199509, 232366, 268169, 312132, 355839, 409902, 465503, 527080, 596443, 668961, 746443, 839830, 937967

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

See the links in A332953 for images of the triangles.

Lars Blomberg, Table of n, a(n) for n = 1..70

Cf. A332953 (regions), A333025 (n-gons), A333026 (vertices), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

A333026 The number of vertices formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

Original entry on oeis.org

3, 6, 16, 45, 111, 230, 448, 769, 1229, 1858, 2860, 4007, 5737, 7724, 10115, 13074, 17172, 21454, 27288, 33332, 40413, 48944, 59594, 70213, 82983, 97608, 113672, 131032, 152986, 174088, 201090, 228295, 258467, 292726, 328080, 365633, 412291, 460834, 512016

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Mar 05 2020

nonn

See the links in A332953 for images of the triangles.

Lars Blomberg, Table of n, a(n) for n = 1..70

Cf. A332953 (regions), A333025 (n-gons), A333027 (edges), A007678, A092867, A331452, A331911, A332357, A332358.

a(16) and beyond from Lars Blomberg, May 26 2020

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:

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

A332358 Main diagonal of A332357.

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A332953 The number of regions formed inside an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A333025 Irregular table read by rows: Take an isosceles triangle with its equal length sides divided into n equal parts with all diagonals drawn, as in A332953. Then T(n,k) = number of k-sided polygons in that figure for k>=3.

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A333027 The number of edges formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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A333026 The number of vertices formed on an isosceles triangle by straight line segments mutually connecting all vertices and all points that divide the two equal length sides into n equal parts; the base of the triangle contains no points other than its vertices.

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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

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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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.

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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_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.

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Maple

Formula

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.

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.