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 A102698 #35 Aug 08 2025 03:26:40
%S A102698 8,80,368,1264,3448,7792,16176,30696,54216,90104,143576,220328,326680,
%T A102698 471232,664648,916344,1241856,1655208,2172584,2812664,3598664,4553800,
%U A102698 5702776,7075264,8705088,10628928,12880056,15496616,18523472,22003808
%N A102698 Number of equilateral triangles with coordinates (x,y,z) in the set {0, 1,...,n}.
%C A102698 Inspired by Problem 25 on the 2005 AMC-12A Mathematics Competition, which asked for a(2).
%H A102698 Eugen J. Ionascu and Rodrigo A. Obando, <a href="/A102698/b102698.txt">Table of n, a(n) for n = 1..100</a>
%H A102698 Ray Chandler and Eugen J. Ionascu, <a href="http://arXiv.org/abs/0710.0708">A characterization of all equilateral triangles in Z^3</a>, arXiv:0710.0708 [math.NT], 2007.
%H A102698 Eugen J. Ionascu, <a href="/A102698/a102698.txt">Maple program</a>
%H A102698 Eugen J. Ionascu, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Ionascu/ionascu2.html">A parametrization of equilateral triangles having integer coordinates</a>, J. Integer Seqs., Vol. 10 (2007), #07.6.7.
%H A102698 Eugen J. Ionascu, <a href="https://www.emis.de/journals/AMUC/_vol-77/_no_1/_ionascu/ionascu.html">Counting all equilateral triangles in {0,1,...,n}^3</a>, Acta Mathematica Universitatis Comenianae, Vol. LXXVII, 1 (2008) p. 129-140.
%H A102698 Rodrigo A. Obando, <a href="/A102698/a102698_b.txt">Mathematica program</a>
%H A102698 Burkard Polster, <a href="https://youtu.be/sDfzCIWpS7Q?t=799">What does this prove? Some of the most gorgeous visual "shrink" proofs ever invented</a>, Mathologer video (2020).
%F A102698 a(n) approximately equals n^4.989; also lim log(a(n))/log(n) exists. - _Eugen J. Ionascu_, Dec 09 2006
%e A102698 a(1) = 8 because in the unit cube, equilateral triangles are formed by cutting off any one of the 8 corners.
%e A102698 a(2) = 80 because there are 8 unit cubes with 8 each, 8 larger triangles (analogous to the 8 in the unit cube, but twice as big) and also 8 triangles of side length sqrt(6).
%p A102698 # See Ionascu link for Maple program.
%t A102698 (* See Obando link for Mathematica program. *)
%Y A102698 Cf. a(n)=8*A103501, A103158 tetrahedra in lattice cube.
%K A102698 nonn
%O A102698 1,1
%A A102698 _Joshua Zucker_, Feb 04 2005
%E A102698 More terms from _Hugo Pfoertner_, Feb 10 2005
%E A102698 Edited by _Ray Chandler_, Sep 15 2007, Jul 27 2010