A057566 Number of collinear triples in a 3 X n rectangular grid.

0, 1, 2, 8, 20, 43, 78, 130, 200, 293, 410, 556, 732, 943, 1190, 1478, 1808, 2185, 2610, 3088, 3620, 4211, 4862, 5578, 6360, 7213, 8138, 9140, 10220, 11383, 12630, 13966, 15392, 16913, 18530, 20248, 22068, 23995, 26030, 28178, 30440, 32821, 35322

Offset: 0

Author

John W. Layman, Oct 04 2000

Keywords

• nonn
• easy

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Second differences give A047264. Third differences are periodic {5, 1, 5, 1, ...} and form A010686. See A000938 for the n X n grid.

Programs

Mathematica

LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 2, 8, 20}, 50] (* Paolo Xausa, Feb 22 2024 *)

Formula

Conjecture: a(n) = 5*floor((2n^3 - 3n^2 - n)/24) + floor((2(n-1)^3 - 3(n-1)^2 - (n-1))/24) + n, which fits all of the listed terms.

From R. J. Mathar, May 23 2010: (Start)

a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5) = n^3/2 - n^2 + n + (1-(-1)^n)/4.

G.f.: x*(1 - x + 4*x^2 + 2*x^3)/((1+x)*(x-1)^4). (End)