A244501 Number of ways to place 3 points on an n X n X n triangular grid so that no pair of them has distance sqrt(3).

1, 8, 55, 248, 820, 2212, 5163, 10815, 20833, 37540, 64067, 104518, 164150, 249568, 368935, 532197, 751323, 1040560, 1416703, 1899380, 2511352, 3278828, 4231795, 5404363, 6835125, 8567532, 10650283, 13137730, 16090298, 19574920, 23665487, 28443313, 33997615

Offset: 2

Author

Heinrich Ludwig, Jun 29 2014

Keywords

• nonn
• easy

Comments

sqrt(3) is the second closest (Euclidean) distance for a pair of points in a triangular grid. For illustration see A244500.

Links

• Heinrich Ludwig, Table of n, a(n) for n = 2..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Cf. A086274, A244500, A244502, A244503.

Programs

Mathematica

CoefficientList[Series[-(6*x^7-17*x^6+14*x^5-6*x^4-4*x^3+20*x^2+x+1) / (x-1)^7, {x, 0, 20}], x] (* Vaclav Kotesovec, Jul 03 2014 after Colin Barker *)

PARI

Vec(-x^2*(6*x^7-17*x^6+14*x^5-6*x^4-4*x^3+20*x^2+x+1)/(x-1)^7 + O(x^100)) \\ Colin Barker, Jun 29 2014

Formula

a(n) = 1/48*n^6 + 1/16*n^5 - 13/16*n^4 + 61/48*n^3 + 247/24*n^2 - 293/6*n + 6 for n >= 3.

G.f.: -x^2*(6*x^7 - 17*x^6 + 14*x^5 - 6*x^4 - 4*x^3 + 20*x^2 + x + 1) / (x-1)^7. - Colin Barker, Jun 29 2014