A244502 Number of ways to place 4 points on an n X n X n triangular grid so that no pair of them has distance sqrt(3).

33, 378, 2190, 9110, 30300, 85563, 213293, 482085, 1006950, 1971185, 3655053, 6472533, 11017505, 18120840, 28919970, 44942618, 68206473, 101336700, 147703280, 211580280, 298329258, 414609113, 568614795, 770347395, 1031918240, 1367889723, 1795655703, 2335864415

Offset: 4

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 = 4..1000
• Index entries for linear recurrences with constant coefficients, signature (9,-36,84,-126,126,-84,36,-9,1).

Crossrefs

Cf. A086274, A244500, A244501, A244503.

Programs

Maple

A244502:=n->`if`(n=4,33,1/384*n^8 + 1/96*n^7 - 13/64*n^6 + 5/48*n^5 + 875/128*n^4 - 2543/96*n^3 - 4141/96*n^2 + 3759/8*n - 837); seq(A244502(n), n=4..30); # Wesley Ivan Hurt, Jun 30 2014

Mathematica

CoefficientList[Series[(10*x^9 - 30*x^8 + 130*x^6 - 333*x^5 + 444*x^4 - 236*x^3 + 24*x^2 - 81*x - 33)/(x - 1)^9, {x, 0, 30}], x] (* Wesley Ivan Hurt, Jun 30 2014 *)

PARI

Vec(x^4*(10*x^9-30*x^8+130*x^6-333*x^5+444*x^4-236*x^3+24*x^2-81*x-33)/(x-1)^9 + O(x^100)) \\ Colin Barker, Jun 29 2014

Formula

a(n) = 1/384*n^8 + 1/96*n^7 - 13/64*n^6 + 5/48*n^5 + 875/128*n^4 - 2543/96*n^3 - 4141/96*n^2 + 3759/8*n - 837, for n >= 5.

G.f.: x^4*(10*x^9 - 30*x^8 + 130*x^6 - 333*x^5 + 444*x^4 - 236*x^3 + 24*x^2 - 81*x - 33) / (x - 1)^9. - Colin Barker, Jun 29 2014