A189414 -id:A189414 - cp's OEIS frontend

A189413 Number of convex quadrilaterals on an n X n grid (or geoboard).

0, 1, 70, 1038, 7398, 35727, 130768, 400116, 1062016, 2531001, 5529310, 11272710, 21639022, 39559591, 69283632, 116910052, 190977408, 303286461, 469431366, 710400658, 1053055398, 1532253131, 2192246528, 3088876728, 4290532688, 5882825641, 7969711934, 10677299074, 14156978846, 18591603883, 24195121104

Offset: 1

Martin Renner, Apr 21 2011

nonn

If four points are chosen at random from an n X n grid, the probability that they form a convex quadrilateral approaches 25/36 as n increases, by Sylvester's Four-Point Theorem (see the link). Thanks to Ed Pegg Jr for this comment. - N. J. A. Sloane, Jun 15 2020

Tom Duff, Table of n, a(n) for n = 1..192
Tom Duff, Data for tables A334708, A334709, A334710, A334711 for grids of size up to 192 X 192.
Nathaniel Johnston, C program for computing terms.
Eric Weisstein's World of Mathematics, Convex Polygon.
Eric Weisstein's World of Mathematics, Quadrilateral.
Eric Weisstein's World of Mathematics, Sylvester's Four-Point Problem.

Cf. A175383, A178256, A181944, A189345, A189412, A189414.

This is the main diagonal of A334711.

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

Terms beyond a(22) from Tom Duff. - N. J. A. Sloane, Jun 23 2020

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

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

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

A189413 Number of convex quadrilaterals on an n X n grid (or geoboard).

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Programs

Python

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions

A189416 Number of parallelograms on an n X n grid.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Links

Crossrefs

Extensions