A189346 -id:A189346 - 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

A189345 Number of ways to choose four points in an n X n grid (or geoplane).

Original entry on oeis.org

0, 1, 126, 1820, 12650, 58905, 211876, 635376, 1663740, 3921225, 8495410, 17178876, 32795126, 59626385, 103962600, 174792640, 284660376, 450710001, 695946630, 1050739900, 1554599970, 2258257001, 3226076876, 4538847600, 6296972500, 8624108025, 11671285626

Offset: 1

Martin Renner, Apr 20 2011

Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1)

Cf. A175383, A189346.

a:= proc(n) binomial(n^2, 4) end: seq(a(n), n=1..35);

makelist(binomial(n^2, 4), n, 1, 56); /* Martin Ettl, Oct 15 2012 */

a(n) = binomial(n^2,4) = (1/24)*n^2*(n^2-1)*(n^2-2)*(n^2-3).

G.f.: x^2*(1+x)*(1+116*x+606*x^2+116*x^3+x^4)/(1-x)^9. - Colin Barker, Jan 19 2012

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

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A189345 Number of ways to choose four points in an n X n grid (or geoplane).

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