A243142 - OEIS

%I #16 Feb 09 2024 11:16:18

%S 0,3,19,75,218,542,1178,2350,4340,7585,12605,20153,31094,46620,68068,

%T 97212,136008,186975,252855,337095,443410,576378,740894,942890,

%U 1188668,1485757,1842113,2267125,2770670,3364280,4060040,4871928,5814544,6904635,8159643,9599427

%N Number of inequivalent (mod D_3) ways to place 3 points on a triangular grid of side n so that they are not vertices of an equilateral triangle of any orientation.

%H Heinrich Ludwig, <a href="/A243142/b243142.txt">Table of n, a(n) for n = 2..1000</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (4,-3,-8,14,0,-14,8,3,-4,1).

%F a(n) = (n^6 + 3*n^5 - 5*n^4 + 6*n^3 - 68*n^2 + 72*n + IF(MOD(n, 2) = 1)*(27*n^2 - 81*n + 45))/288.

%F G.f.: x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3) / ((x-1)^7*(x+1)^3). - _Colin Barker_, May 30 2014

%t Drop[CoefficientList[Series[x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3) / ((x-1)^7*(x+1)^3), {x, 0, 40}], x],2] (* _Vaclav Kotesovec_, May 31 2014 after _Colin Barker_ *)

%o (PARI) concat(0, Vec(x^3*(2*x^5-5*x^4+x^3-8*x^2-7*x-3)/((x-1)^7*(x+1)^3) + O(x^100))) \\ _Colin Barker_, May 30 2014

%Y Cf. A243141, A240440, A001399, A227327, A243143, A243144.

%K nonn,easy

%O 2,2

%A _Heinrich Ludwig_, May 30 2014