A334705 -id:A334705 - cp's OEIS frontend

A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

0, 0, 0, 1, 2, 8, 4, 8, 20, 44, 10, 20, 43, 84, 152, 20, 40, 78, 140, 240, 372, 35, 70, 130, 224, 369, 558, 824, 56, 112, 200, 332, 528, 780, 1132, 1544, 84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448

Offset: 1

N. J. A. Sloane, Jun 13 2020

It follows from the definitions that T(n,k) + A334705(n,k) = A334703(n,k) for 1 <= k <= n.

			Triangle begins:
0,
0, 0,
1, 2, 8,
4, 8, 20, 44,
10, 20, 43, 84, 152,
20, 40, 78, 140, 240, 372,
35, 70, 130, 224, 369, 558, 824,
56, 112, 200, 332, 528, 780, 1132, 1544,
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712,
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448,
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992,
...
This is the lower half of a symmetric array. The full symmetric array begins:
0, 0, 1, 4, 10, 20, 35, 56, 84, 120, 165, 220, ...
0, 0, 2, 8, 20, 40, 70, 112, 168, 240, 330, 440, ...
1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, ...
4, 8, 20, 44, 84, 140, 224, 332, 472, 648, 864, 1120, ...
10, 20, 43, 84, 152, 240, 369, 528, 734, 988, 1295, 1652, ...
20, 40, 78, 140, 240, 372, 558, 780, 1064, 1408, 1826, 2304, ...
35, 70, 130, 224, 369, 558, 824, 1132, 1519, 1982, 2542, 3172, ...
56, 112, 200, 332, 528, 780, 1132, 1544, 2052, 2652, 3372, 4172, ...
84, 168, 293, 472, 734, 1064, 1519, 2052, 2712, 3480, 4393, 5396, ...
120, 240, 410, 648, 988, 1408, 1982, 2652, 3480, 4448, 5586, 6824, ...
165, 330, 556, 864, 1295, 1826, 2542, 3372, 4393, 5586, 6992, 8508, ...
220, 440, 732, 1120, 1652, 2304, 3172, 4172, 5396, 6824, 8508, 10332, ...
...

Tom Duff, Values of T(n,k) for grids of size up to 50 X 50

This is a companion to the triangles A334703 and A334705.

Rows (or columns) 1,2,3,4 of the full array are A000292, A007290, A057566, A334706. The main diagonal is A000938.

Rows 6 onwards from Tom Duff. - N. J. A. Sloane, Jun 19 2020

A334708 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four collinear points from an n X k grid.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 5, 2, 0, 2, 5, 15, 10, 3, 3, 10, 15, 35, 30, 15, 10, 15, 30, 35, 70, 70, 45, 29, 29, 45, 70, 70, 126, 140, 105, 72, 64, 72, 105, 140, 126, 210, 252, 210, 157, 129, 129, 157, 210, 252, 210, 330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 1, 5, 15, 35, 70, 126, 210, 330, 495, ...
0, 0, 0, 2, 10, 30, 70, 140, 252, 420, 660, 990, ...
0, 0, 0, 3, 15, 45, 105, 210, 378, 630, 990, 1485, ...
1, 2, 3, 10, 29, 72, 157, 302, 531, 874, 1361, 2028, ...
5, 10, 15, 29, 64, 129, 248, 442, 747, 1196, 1825, 2679, ...
15, 30, 45, 72, 129, 234, 405, 666, 1065, 1638, 2439, 3510, ...
35, 70, 105, 157, 248, 405, 660, 1020, 1545, 2276, 3283, 4605, ...
70, 140, 210, 302, 442, 666, 1020, 1524, 2220, 3154, 4412, 6030, ...
126, 252, 378, 531, 747, 1065, 1545, 2220, 3156, 4362, 5940, 7923, ...
210, 420, 630, 874, 1196, 1638, 2276, 3154, 4362, 5928, 7914, 10350, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
1, 0, 0, 1
5, 2, 0, 2, 5
15, 10, 3, 3, 10, 15
35, 30, 15, 10, 15, 30, 35
70, 70, 45, 29, 29, 45, 70, 70
126, 140, 105, 72, 64, 72, 105, 140, 126
210, 252, 210, 157, 129, 129, 157, 210, 252, 210
330, 420, 378, 302, 248, 234, 248, 302, 378, 420, 330
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A178256.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334709 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is strictly inside the triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 48, 48, 0, 0, 0, 0, 144, 240, 144, 0, 0, 0, 0, 348, 716, 716, 348, 0, 0, 0, 0, 700, 1712, 2100, 1712, 700, 0, 0, 0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0, 0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0, 0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 8, 48, 144, 348, 700, 1280, 2144, 3400, 5120, 7440, ...
0, 0, 48, 240, 716, 1712, 3404, 6176, 10336, 16288, 24480, 35504, ...
0, 0, 144, 716, 2100, 4984, 9900, 17936, 29924, 47080, 70700, 102460, ...
0, 0, 348, 1712, 4984, 11604, 22936, 41372, 68844, 108132, 161964, 234228, ...
0, 0, 700, 3404, 9900, 22936, 45184, 81320, 135192, 212152, 317492, 458812, ...
0, 0, 1280, 6176, 17936, 41372, 81320, 145648, 241544, 378400, 565636, 816520, ...
0, 0, 2144, 10336, 29924, 68844, 135192, 241544, 399656, 625232, 933808, 1346928, ...
0, 0, 3400, 16288, 47080, 108132, 212152, 378400, 625232, 976552, 1457172, 2100112, ...
...
The initial antidiagonals are:
0,
0, 0,
0, 0, 0,
0, 0, 0, 0,
0, 0, 8, 0, 0,
0, 0, 48, 48, 0, 0,
0, 0, 144, 240, 144, 0, 0,
0, 0, 348, 716, 716, 348, 0, 0,
0, 0, 700, 1712, 2100, 1712, 700, 0, 0,
0, 0, 1280, 3404, 4984, 4984, 3404, 1280, 0, 0,
0, 0, 2144, 6176, 9900, 11604, 9900, 6176, 2144, 0, 0,
0, 0, 3400, 10336, 17936, 22936, 22936, 17936, 10336, 3400, 0, 0,
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A334712.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334710 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 6, 6, 0, 0, 32, 48, 32, 0, 0, 100, 168, 168, 100, 0, 0, 240, 456, 532, 456, 240, 0, 0, 490, 990, 1312, 1312, 990, 490, 0, 0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0, 0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0, 0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 0, 6, 32, 100, 240, 490, 896, 1512, 2400, 3630, 5280, ...
0, 6, 48, 168, 456, 990, 1920, 3360, 5520, 8550, 12720, 18216, ...
0, 32, 168, 532, 1312, 2652, 4972, 8420, 13452, 20480, 29980, 42288, ...
0, 100, 456, 1312, 3088, 5964, 10816, 17768, 27840, 41652, 60040, 83448, ...
0, 240, 990, 2652, 5964, 11340, 20142, 32436, 50004, 73704, 105282, 144936, ...
0, 490, 1920, 4972, 10816, 20142, 35264, 55916, 84960, 123690, 174976, 238512, ...
0, 896, 3360, 8420, 17768, 32436, 55916, 88088, 132708, 191588, 268972, 363876, ...
0, 1512, 5520, 13452, 27840, 50004, 84960, 132708, 198912, 285312, 397968, 534888, ...
0, 2400, 8550, 20480, 41652, 73704, 123690, 191588, 285312, 407744, 566046, 757008, ...
...
The initial antidiagonals are:
0
0, 0
0, 0, 0
0, 6, 6, 0
0, 32, 48, 32, 0
0, 100, 168, 168, 100, 0
0, 240, 456, 532, 456, 240, 0
0, 490, 990, 1312, 1312, 990, 490, 0
0, 896, 1920, 2652, 3088, 2652, 1920, 896, 0
0, 1512, 3360, 4972, 5964, 5964, 4972, 3360, 1512, 0
0, 2400, 5520, 8420, 10816, 11340, 10816, 8420, 5520, 2400, 0
0, 3630, 8550, 13452, 17768, 20142, 20142, 17768, 13452, 8550, 3630, 0
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192

The main diagonal is A334713.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334711 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that they form a convex quadrilateral.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 9, 9, 0, 0, 36, 70, 36, 0, 0, 100, 276, 276, 100, 0, 0, 225, 750, 1038, 750, 225, 0, 0, 441, 1677, 2788, 2788, 1677, 441, 0, 0, 784, 3260, 6190, 7398, 6190, 3260, 784, 0, 0, 1296, 5776, 11942, 16328, 16328, 11942, 5776, 1296, 0, 0, 2025, 9508, 21062, 31396, 35727, 31396, 21062, 9508, 2025, 0

Offset: 1

N. J. A. Sloane, Jun 15 2020

Computed by Tom Duff, Jun 15 2020

For the limiting probability that the four points form a convex quadrilateral when n and k are large, see the link to Sylvester's Four-Point Problem. Thanks to Ed Pegg Jr for this comment.

			The initial rows of the array are:
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
0, 1, 9, 36, 100, 225, 441, 784, 1296, 2025, 3025, 4356, ...
0, 9, 70, 276, 750, 1677, 3260, 5776, 9508, 14825, 22090, 31764, ...
0, 36, 276, 1038, 2788, 6190, 11942, 21062, 34586, 53748, 79930, 114760, ...
0, 100, 750, 2788, 7398, 16328, 31396, 55244, 90484, 140372, 208490, 299048, ...
0, 225, 1677, 6190, 16328, 35727, 68447, 120106, 196338, 304161, 451035, 646116, ...
0, 441, 3260, 11942, 31396, 68447, 130768, 229034, 373968, 578777, 857524, 1227572, ...
0, 784, 5776, 21062, 55244, 120106, 229034, 400116, 652318, 1008438, 1492870, 2135534, ...
0, 1296, 9508, 34586, 90484, 196338, 373968, 652318, 1062016, 1640284, 2426660, 3469356, ...
0, 2025, 14825, 53748, 140372, 304161, 578777, 1008438, 1640284, 2531001, 3742053, 5347100, ...
...
The initial antidiagonals are:
0,
0, 0,
0, 1, 0,
0, 9, 9, 0,
0, 36, 70, 36, 0,
0, 100, 276, 276, 100, 0,
0, 225, 750, 1038, 750, 225, 0,
0, 441, 1677, 2788, 2788, 1677, 441, 0,
0, 784, 3260, 6190, 7398, 6190, 3260, 784, 0,
0, 1296, 5776, 11942, 16328, 16328, 11942, 5776, 1296, 0,
0, 2025, 9508, 21062, 31396, 35727, 31396, 21062, 9508, 2025, 0,
0, 3025, 14825, 34586, 55244, 68447, 68447, 55244, 34586, 14825, 3025, 0,
...

Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192
Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

The main diagonal is A189413.
Triangles A334708, A334709, A334710, A334711 give the counts for the four possible arrangements of four points.
For three points there are just two possible arrangements: see A334704 and A334705.

A334703 Triangle read by rows: T(n,k) = binomial(n*k,3) (0 <= k <= n).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 0, 1, 20, 84, 0, 4, 56, 220, 560, 0, 10, 120, 455, 1140, 2300, 0, 20, 220, 816, 2024, 4060, 7140, 0, 35, 364, 1330, 3276, 6545, 11480, 18424, 0, 56, 560, 2024, 4960, 9880, 17296, 27720, 41664, 0, 84, 816, 2925, 7140, 14190, 24804, 39711, 59640, 85320

Offset: 0

N. J. A. Sloane, Jun 13 2020

			Triangle begins:
[0]
[0, 0]
[0, 0, 4]
[0, 1, 20, 84]
[0, 4, 56, 220, 560]
[0, 10, 120, 455, 1140, 2300]
[0, 20, 220, 816, 2024, 4060, 7140]
[0, 35, 364, 1330, 3276, 6545, 11480, 18424]
[0, 56, 560, 2024, 4960, 9880, 17296, 27720, 41664]
[0, 84, 816, 2925, 7140, 14190, 24804, 39711, 59640, 85320]
...

See A334702 for another version.
This is a companion to the triangles A334704 and A334705.
Cf. A007318.

A334707 Number of non-collinear triples in a 5 X n rectangular grid.

Original entry on oeis.org

0, 100, 412, 1056, 2148, 3820, 6176, 9352, 13456, 18612, 24940, 32568, 41596, 52164, 64384, 78376, 94256, 112156, 132180, 154464, 179116, 206260, 236016, 268512, 303848, 342164, 383572, 428192, 476140, 527548, 582520, 641192, 703672, 770084, 840548, 915192, 994116

Offset: 1

N. J. A. Sloane, Jun 13 2020

nonn

Giovanni Resta, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-1,-1,1,-1,2,-1).

A row of A334705.

From Stefano Spezia, Jun 20 2020: (Start)
G.f.: 4*x*(25 + 53*x + 83*x^2 + 87*x^3 + 67*x^4 + 35*x^5 + 10*x^6)/((1 - x)^4*(1 + 2*x + 3*x^2 + 3*x^3 + 2*x^4 + x^5)).
a(n) = 2*a(n-1) - a(n-2) + a(n-3) - a(n-4) - a(n-5) + a(n-6) - a(n-7) + 2*a(n-8) - a(n-9) for n > 9. (End)

Terms a(6) and beyond from Giovanni Resta, Jun 20 2020

cp's OEIS Frontend

A334704 Triangle read by rows: T(n,k) (1 <= k <= n) = number of ways to choose three collinear points from an n X k grid of points.

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A334708 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four collinear points from an n X k grid.

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A334709 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is strictly inside the triangle.

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A334710 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that three of them form a triangle of nonzero area and the extra point is on one of the edges of the triangle.

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A334711 Array read by antidiagonals: T(n,k) (n>=1, k>=1) = number of ways to select four points from an n X k grid so that they form a convex quadrilateral.

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A334703 Triangle read by rows: T(n,k) = binomial(n*k,3) (0 <= k <= n).

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Examples

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A334707 Number of non-collinear triples in a 5 X n rectangular grid.

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