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A243213 Number of ways to place 4 points on a triangular grid of side length n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.

%I A243213 #17 Jun 13 2015 00:55:03
%S A243213 3,128,1062,5160,18591,55113,142005,329045,701160,1395975,2626953,
%T A243213 4713723,8120322,13503350,21770766,34153758,52292385,78337890,
%U A243213 115072320,166048850,235753353,329791143,455099307,620189115,835418766,1113301553,1468849515,1919958285
%N A243213 Number of ways to place 4 points on a triangular grid of side length n so that no three of them are vertices of an equilateral triangle with sides parallel to the grid.
%H A243213 Heinrich Ludwig, <a href="/A243213/b243213.txt">Table of n, a(n) for n = 3..1000</a>
%H A243213 <a href="/index/Rec#order_12">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1)
%F A243213 a(n) = (n^8 + 4*n^7 - 6*n^6 - 80*n^5 - 15*n^4 + 532*n^3 - 244*n^2 - 432*n)/384 + IF(MOD(n, 2) = 1)*(-n^2 - n + 12)/16.
%F A243213 G.f.: x^3*(7*x^7-33*x^6-15*x^5-38*x^4-318*x^3-330*x^2-110*x-3) / ((x-1)^9*(x+1)^3). - _Colin Barker_, Jun 09 2014
%e A243213 There are exactly a(3) = 3 ways to place 4 points (x) on a 3X3X3 grid, no three of them being vertices of an equilateral triangle:
%e A243213       .            x            x
%e A243213      x x          . x          x .
%e A243213     x . x        x x .        . x x
%o A243213 (PARI) Vec(x^3*(7*x^7-33*x^6-15*x^5-38*x^4-318*x^3-330*x^2-110*x-3)/((x-1)^9*(x+1)^3) + O(x^100)) \\ _Colin Barker_, Jun 09 2014
%Y A243213 Cf. A243211, A243209, A000217, A050534, A243212, A243214.
%K A243213 nonn,easy
%O A243213 3,1
%A A243213 _Heinrich Ludwig_, Jun 09 2014