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 A008893 #43 Aug 16 2025 23:56:37
%S A008893 0,8,66,258,710,1590,3108,5516,9108,14220,21230,30558,42666,58058,
%T A008893 77280,100920,129608,164016,204858,252890,308910,373758,448316,533508,
%U A008893 630300,739700,862758,1000566,1154258,1325010,1514040,1722608,1952016,2203608,2478770
%N A008893 Number of equilateral triangles formed by triples of points taken from a hexagonal chunk of side n in the hexagonal lattice.
%C A008893 The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice. Here we consider a hexagonal chunk of the lattice in which each bounding edge contains n+1 points.
%H A008893 Nathaniel Johnston, <a href="/A008893/b008893.txt">Table of n, a(n) for n = 0..10000</a>
%H A008893 Gabriele Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/A2.html">Home page for hexagonal (or triangular) lattice A2</a>.
%H A008893 N. J. A. Sloane, <a href="/A008893/a008893.jpg">Illustration for a(1)=8.</a> [The drawing was made for a different offset, so it says a(2)=8.]
%H A008893 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F A008893 a(n) = n*(n+1)*(7*n^2+7*n+2)/4.
%F A008893 G.f.: -2*x*(4*x^2+13*x+4)/(x-1)^5 [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 10 2009]
%F A008893 From _Elmo R. Oliveira_, Aug 15 2025: (Start)
%F A008893 E.g.f.: exp(x)*x*(2 + x)*(16 + 42*x + 7*x^2)/4.
%F A008893 a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5).
%F A008893 a(n) = 2*A152041(n). (End)
%t A008893 A008893[n_] := n*(n + 1)*(7*n*(n + 1) + 2)/4; Array[A008893, 50, 0] (* or *)
%t A008893 LinearRecurrence[{5, -10, 10, -5, 1}, {0, 8, 66, 258, 710}, 50] (* _Paolo Xausa_, Aug 16 2025 *)
%o A008893 (Maxima) A008893(n):=n*(n+1)*(7*n^2+7*n+2)/4$
%o A008893 makelist(A008893(n),n,0,30); /* _Martin Ettl_, Nov 03 2012 */
%Y A008893 Cf. A045949, A152041.
%K A008893 nonn,easy
%O A008893 0,2
%A A008893 _N. J. A. Sloane_, _R. K. Guy_
%E A008893 Edited May 29 2012 by _N. J. A. Sloane_, May 29 2012