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.
3, 128, 1062, 5160, 18591, 55113, 142005, 329045, 701160, 1395975, 2626953, 4713723, 8120322, 13503350, 21770766, 34153758, 52292385, 78337890, 115072320, 166048850, 235753353, 329791143, 455099307, 620189115, 835418766, 1113301553, 1468849515, 1919958285
Offset: 3
Examples
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:
. x x
x x . x x .
x . x x x . . x x
Links
- Heinrich Ludwig, Table of n, a(n) for n = 3..1000
- Index entries for linear recurrences with constant coefficients, signature (6,-12,2,27,-36,0,36,-27,-2,12,-6,1)
Programs
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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
Formula
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.
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