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

2,2

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).

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

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 *)

Prog

(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

Crossrefs

Cf. A086274, A244500, A244502, A244503.

Sequence in context: A116885 A179407 A373116 * A026994 A110184 A013698

Adjacent sequences: A244498 A244499 A244500 * A244502 A244503 A244504

Keyword

nonn,easy

Author

Heinrich Ludwig, Jun 29 2014

Status

approved