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

0,3

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).

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)

Mathematica

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

Crossrefs

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

Sequence in context: A048096 A072250 A220908 * A333642 A009303 A096586

Adjacent sequences: A057563 A057564 A057565 * A057567 A057568 A057569

Keyword

nonn,easy

Author

John W. Layman, Oct 04 2000

Status

approved