This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A057566 #27 Feb 22 2024 10:43:45
%S A057566 0,1,2,8,20,43,78,130,200,293,410,556,732,943,1190,1478,1808,2185,
%T A057566 2610,3088,3620,4211,4862,5578,6360,7213,8138,9140,10220,11383,12630,
%U A057566 13966,15392,16913,18530,20248,22068,23995,26030,28178,30440,32821,35322
%N A057566 Number of collinear triples in a 3 X n rectangular grid.
%H A057566 Paolo Xausa, <a href="/A057566/b057566.txt">Table of n, a(n) for n = 0..10000</a>
%H A057566 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A057566 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.
%F A057566 From _R. J. Mathar_, May 23 2010: (Start)
%F A057566 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.
%F A057566 G.f.: x*(1 - x + 4*x^2 + 2*x^3)/((1+x)*(x-1)^4). (End)
%t A057566 LinearRecurrence[{3, -2, -2, 3, -1}, {0, 1, 2, 8, 20}, 50] (* _Paolo Xausa_, Feb 22 2024 *)
%Y A057566 Second differences give A047264. Third differences are periodic {5, 1, 5, 1, ...} and form A010686. See A000938 for the n X n grid.
%K A057566 nonn,easy
%O A057566 0,3
%A A057566 _John W. Layman_, Oct 04 2000