A240441 Number of ways to place 4 points on a triangular grid of side n so that no three of these points are vertices of an equilateral triangle of any orientation.

0, 0, 3, 114, 969, 4773, 17415, 52125, 135375, 315675, 676200, 1352085, 2553558, 4595934, 7937874, 13229118, 21369330, 33579450, 51487425, 77229900, 113571975, 164046795, 233117313, 326362179, 450688329, 614572413, 828333870, 1104441975, 1457859900, 1906428300

Offset: 1

Author

Heinrich Ludwig, Apr 05 2014

Keywords

• nonn
• easy

Links

• Heinrich Ludwig, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-19,21,6,-42,42,-6,-21,19,-7,1).

Crossrefs

Cf. A240439, A240440, A240442.

Programs

Mathematica

Table[(n^8+4*n^7-14*n^6-56*n^5+61*n^4+220*n^3-84*n^2-240*n)/384 +If[EvenQ[n],0,(6*n+3)/32],{n,1,20}] (* Vaclav Kotesovec, Apr 05 2014 after Heinrich Ludwig *)

PARI

concat([0,0], Vec(-3*x^3*(x^4+31*x^3+76*x^2+31*x+1)/((x-1)^9*(x+1)^2) + O(x^100))) \\ Colin Barker, Apr 05 2014

Formula

a(n) = (n^8 + 4*n^7 - 14*n^6 - 56*n^5 + 61*n^4 + 220*n^3 - 84*n^2 - 240*n)/384 + IF(MOD(n, 2) = 1)*(6*n + 3)/32.

G.f.: -3*x^3*(x^4+31*x^3+76*x^2+31*x+1) / ((x-1)^9*(x+1)^2). - Colin Barker, Apr 05 2014