A189413 -id:A189413 - cp's OEIS frontend

A175383 Number of complete quadrangles on an n X n grid (or geoplane).

0, 1, 78, 1278, 9498, 47331, 175952, 545764, 1461672, 3507553, 7701638, 15773526, 30375194, 55695587, 97777392, 165310348, 270478344, 430196181, 666685134, 1010083690, 1498720098, 2182544223

Offset: 1

Martin Renner, Apr 19 2011

nonn

A complete quadrangle is a set of four points, no three collinear, and the six lines which join them.

Number of ways to arrange 4 indistinguishable points on an n X n square grid so that no three points are collinear at any angle. Column 4 of A194193. - R. H. Hardin, Aug 18 2011

			From _R. H. Hardin_, Aug 18 2011: (Start)
Some solutions for 3 X 3:
  0 1 1   1 1 0   1 0 1   0 1 1   0 0 0   1 1 0   1 1 0
  1 0 0   0 0 0   1 0 0   1 1 0   1 1 0   0 0 1   1 0 0
  1 0 0   1 0 1   0 0 1   0 0 0   0 1 1   0 1 0   0 1 0
(End)

R. H. Hardin, Table of n, a(n) for n = 1..55
Eric Weisstein's World of Mathematics, Complete Quadrangle.

a(n) = A189345(n) - A189346(n) - A178256(n).

a(n) = (1/3)*A189412(n) + A189413(n).

a(6)-a(22) from Nathaniel Johnston, Apr 25 2011

a(7)-a(22) corrected by Nathaniel Johnston, based on another correction by Michal Forišek, Sep 06 2011

A189412 Number of concave quadrilaterals on an n X n grid (or geoboard).

Original entry on oeis.org

0, 0, 24, 720, 6300, 34812, 135552, 436944, 1198968, 2929656, 6516984, 13502448, 26208516, 48407988, 85481280, 145200888, 238502808, 380729160, 591761304, 899049096, 1336994100, 1950873276, 2798226336, 3952174032, 5500597632, 7555866072, 10253438688

Offset: 1

Martin Renner, Apr 21 2011

nonn

Michal Forišek, Table of n, a(n) for n = 1..50
Eric Weisstein's World of Mathematics, Concave Polygon.
Eric Weisstein's World of Mathematics, Quadrilateral.

Cf. A175383, A189345, A189413, A189414.

def gcd(x, y):
  x, y = abs(x), abs(y)
  while y: x, y = y, x%y
  return x
def concave(N):
  V = [ (r, c) for r in range(-N+1, N) for c in range(N) if (c>0 or r>0) ]
  answer = 0
  for i in range(len(V)):
    for j in range(i):
      r1, c1, r2, c2 = V[i]+V[j]
      rr, cr, ta = N-max(r1, r2, 0)+min(r1, r2, 0), N-max(c1, c2), abs(r1*c2-r2*c1)
      if rr>0 and cr>0 and ta>0:
        answer += 3*rr*cr*(ta+2-gcd(r1, c1)-gcd(r2, c2)-gcd(r1-r2, c1-c2))/2
  return answer
for N in range(1, 28):
    print(int(concave(N)), end=', ')

a(6)-a(22) from Nathaniel Johnston, Apr 25 2011
Terms a(7)-a(22) corrected by Michal Forisek, Sep 06 2011
Terms a(23)-a(50) added by Michal Forisek, Sep 06 2011

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.

A189414 Number of quadrilaterals on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 94, 1758, 13698, 70539, 266320, 837060, 2260984, 5460657, 12046294, 24775158, 47847538, 87967579, 154764912, 262110940, 429480216, 684015621, 1061192670, 1609449754, 2390049498, 3483126407

Offset: 1

Martin Renner, Apr 21 2011

nonn

Eric Weisstein's World of Mathematics, Quadrilateral.

Cf. A175383, A189345, A189412, A189413.

a(n) = A189412(n) + A189413(n).

a(6) - a(22) from Nathaniel Johnston, Apr 25 2011

a(7) - a(22) corrected by Michal Forisek, Sep 06 2011

A189418 Number of rhombi on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 6, 22, 66, 151, 312, 564, 984, 1601, 2478, 3622, 5242, 7271, 9856, 13124, 17296, 22229, 28286, 35306, 43850, 53891, 65520, 78624, 94272, 111977, 131990, 154514, 180290, 208611, 240840, 276032, 315720, 359497, 407470, 460078, 519018, 582447, 651232, 725820, 808416

Offset: 1

Martin Renner, Apr 21 2011

nonn

Ray Chandler, Table of n, a(n) for n = 1..1000 (first 92 terms from R. H. Hardin)
Nathaniel Johnston, C program for computing terms
Eric Weisstein's World of Mathematics, Rhombus

Cf. A181947, A189413, A189414.

a(6)-a(28) from Nathaniel Johnston, Apr 23 2011

Terms beyond a(28) by R. H. Hardin, May 04 2011

A181944 Number of convex quadrilaterals, distinct up to congruence, on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 12, 89, 407, 1413, 3894, 9431, 20212, 39847, 73177, 127582, 211012, 337186, 519594, 777447, 1134269, 1620415, 2264873, 3114709, 4209184, 5609209, 7378581, 9594611, 12326333, 15688198, 19779188, 24721601, 30646522, 37727553, 46093734, 55983150, 67558997

Offset: 1

Martin Renner, Apr 03 2012

nonn

			a(1) = 0 because the 1 X 1 grid has no quadrilaterals.
a(2) = 1 because the 2 X 2 grid has one quadrilateral.
a(3) = 9 because the 3 X 3 grid has 12 congruence classes of quadrilaterals, out of 70 quadrilaterals total:
+-------+-------+-------+-------+
| . . . | . o . | . . . | . o . |
| o o . | o . . | o . o | o . . |
| o o . | o o . | o . o | o . o |
+-------+-------+-------+-------+
| . . o | o . o | . o . | . o . |
| o . . | . . . | o o . | o . o |
| o . o | o . o | o . . | o . . |
+-------+-------+-------+-------+
| . o o | . . o | . o . | . . o |
| o . . | o . o | o . o | o . . |
| o . . | o . . | . o . | o o . |
+-------+-------+-------+-------+

Lucas A. Brown, Python program.
Eric Weisstein's World of Mathematics, Convex Polygon.
Eric Weisstein's World of Mathematics, Quadrilateral.

Cf. A189413.

a(7)-a(33) from Lucas A. Brown, Feb 06 2024

A189416 Number of parallelograms on an n X n grid.

Original entry on oeis.org

0, 1, 22, 158, 674, 2159, 5664, 13004, 26904, 51401, 92094, 156710, 255090, 400359, 608656, 900100, 1299336, 1836461, 2546550, 3472162, 4661898, 6173123, 8071952, 10434600, 13346080, 16905033, 21221558, 26419338, 32636098, 40027283, 48761448

Offset: 1

Martin Renner, Apr 21 2011

nonn

Nathaniel Johnston, C program for computing terms
Eric Weisstein's World of Mathematics, Parallelogram

Cf. A186434, A189413, A189414.

a[n_] := Sum[(n-a)*(n-b)*(2*a*b - GCD[a, b]), {a, 1, n-1}, {b, 1, n-1}];
Array[a, 31] (* Jean-François Alcover, Oct 08 2017, translated from PARI *)

a(n) = sum(a=1, n-1, sum(b=1, n-1, (n-a)*(n-b)*(2*a*b - gcd(a,b)) )); \\ Andrew Howroyd, Sep 19 2017

a(n) = Sum_{a=1..n-1} Sum_{b=1..n-1} (n-a)*(n-b)*(2*a*b - gcd(a,b)). - Andrew Howroyd, Sep 19 2017

a(6)-a(31) from Nathaniel Johnston, Apr 24 2011

A189417 Number of kites on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 10, 58, 222, 631, 1584, 3340, 6504, 11697, 19978, 31922, 49822, 74167, 107672, 152484, 211944, 286725, 383578, 502262, 651526, 833979, 1056104, 1318104, 1637336, 2011577, 2452634, 2965902, 3568086, 4253755, 5055448, 5960480, 6999104, 8173985, 9503674, 10994202

Offset: 1

Martin Renner, Apr 21 2011

nonn

Only convex kites are counted, not concave kites (sometimes called darts or arrowheads).

Lucas A. Brown, Python program.
Nathaniel Johnston, C program for computing terms
Eric Weisstein's World of Mathematics, Kite.

Cf. A181946, A189413, A189414.

a(7)-a(29) from Nathaniel Johnston, Apr 27 2011

a(30)-a(36) from Lucas A. Brown, Feb 09 2024

A189415 Number of trapezoids on an n X n grid (or geoboard).

Original entry on oeis.org

0, 1, 50, 490, 2618, 9519, 28432, 70796, 157912, 321161, 610482, 1082570, 1848362, 3003015, 4716792, 7204604, 10730528, 15530189, 22093410, 30723078, 42146178, 56981411, 75952240, 99685104, 129757248

Offset: 1

Martin Renner, Apr 21 2011

nonn

Eric Weisstein's World of Mathematics, Trapezoid.
Nathaniel Johnston, C program for computing terms.

Cf. A181945, A186434, A189413, A189414, A189416, A189418.

a(6)-a(25) from Nathaniel Johnston, Apr 25 2011

cp's OEIS Frontend

A175383 Number of complete quadrangles on an n X n grid (or geoplane).

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A189412 Number of concave quadrilaterals on an n X n grid (or geoboard).

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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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A189414 Number of quadrilaterals on an n X n grid (or geoboard).

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A189418 Number of rhombi on an n X n grid (or geoboard).

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A181944 Number of convex quadrilaterals, distinct up to congruence, on an n X n grid (or geoboard).

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A189416 Number of parallelograms on an n X n grid.

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A189417 Number of kites on an n X n grid (or geoboard).

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A189415 Number of trapezoids on an n X n grid (or geoboard).

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