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 A307011 #29 Jul 25 2021 02:17:11 %S A307011 0,1,0,-1,-1,0,1,2,2,1,0,-1,-2,-2,-2,-1,0,1,2,3,3,3,2,1,0,-1,-2,-3,-3, %T A307011 -3,-3,-2,-1,0,1,2,3,4,4,4,4,3,2,1,0,-1,-2,-3,-4,-4,-4,-4,-4,-3,-2,-1, %U A307011 0,1,2,3,4,5,5,5,5,5,4,3,2,1,0,-1,-2,-3,-4,-5 %N A307011 First coordinate in a redundant hexagonal coordinate system of the points of a counterclockwise spiral on an hexagonal grid. Second and third coordinates are given in A307012 and A345978. %C A307011 From _Peter Munn_, Jul 22 2021: (Start) %C A307011 The points of the spiral are equally the points of a hexagonal lattice, the points of an isometric (triangular) grid and the center points of the cells of a honeycomb (regular hexagonal tiling or grid). The coordinate system can be described using 3 axes that pass through spiral point 0 and one of points 1, 2 or 3. Along each axis, one of the coordinates is 0. %C A307011 a(n) is the signed distance from spiral point n to the axis that passes through point 2. The distance is measured along either of the lines through point n that are parallel to one of the other 2 axes and the sign is such that point 1 has positive distance. %C A307011 This coordinate can be paired with either of the other coordinates to form oblique coordinates as described in A307012. Alternatively, all 3 coordinates can be used together, symmetrically, as described in A345978. %C A307011 There is a negated variant of the 3rd coordinate, which is the conventional sense of this coordinate for specifying (with the 2nd coordinate) the Eisenstein integers that can be the points of the spiral when it is embedded in the complex plane. See A307013. %C A307011 (End) %H A307011 Hugo Pfoertner, <a href="/A307011/b307011.txt">Table of n, a(n) for n = 0..10034</a> %H A307011 Margherita Barile, <a href="https://mathworld.wolfram.com/ObliqueCoordinates.html">Oblique Coordinates</a>, entry in Eric Weisstein's World of Mathematics. %H A307011 HandWiki, <a href="https://handwiki.org/wiki/Hexagonal_lattice">Hexagonal Lattice</a>. %H A307011 Peter Munn, <a href="/A307011/a307011.png">Illustration of signed distance of spiral points</a>. %H A307011 Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A307011&name2=A307012&tform1=untransformed&tform2=untransformed&shift=0&radiop1=xy&drawlines=true">Illustration of A307012 vs A307011</a>, spiral. %H A307011 Hugo Pfoertner, <a href="https://oeis.org/plot2a?name1=A307011&name2=A345978&tform1=untransformed&tform2=untransformed&shift=0&radiop1=xy&drawlines=true">Illustration of A345978 vs A307011</a>, spiral. %H A307011 Wikipedia, <a href="https://en.m.wikipedia.org/wiki/Signed_distance_function">Signed distance function</a>. %o A307011 (PARI) r=-1;d=-1;print1(m=0,", ");for(k=0,8,for(j=1,r,print1(s,", "));if(k%2,,m++;r++);for(j=-m,m+1,if(d*j>=-m,print1(s=d*j,", ")));d=-d) %Y A307011 Cf. A307012, A307013, A307014, A307016, A307017, A328818, A345978. %Y A307011 Numbers on the spokes of the spiral: A000567, A028896, A033428, A045944, A049450, A049451. %Y A307011 Positions on the spiral that correspond to Eisenstein primes: A345435. %K A307011 sign,look %O A307011 0,8 %A A307011 _Hugo Pfoertner_, Mar 19 2019 %E A307011 Name revised by _Peter Munn_, Jul 08 2021