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 A346723 #23 Feb 16 2025 08:34:02 %S A346723 0,2,3,5,6,7,8,9,11,13,14,15,17,19,20,21,22,23,25,26,28,29,30,31,32, %T A346723 34,35,39,41,51,53,91,101,109,119,128,129,130,132,134,136,137,138,141, %U A346723 143,144,146,149,150,151,153,155,157,158,159,162,164,165,167,171,175,177 %N A346723 Use the cells of a hexagonal grid to represent the algebraic integers in the integer ring of Q(sqrt(-19)) as explained in the comments. Number the cells along the counterclockwise hexagonal spiral that starts with cells 0 and 1 representing integers 0 and 1. List the cells that represent 0 or a prime in the ring. %C A346723 In this entry we use "rational integers" to refer to integers in their usual sense as whole numbers - they form a subset of the algebraic integers that form the ring, which we denote "R". %C A346723 The algebraic integers in R (the elements of R) are specifically quadratic integers of the form z = x + y*sqrt(-19) or z = (x+0.5) + (y+0.5)*sqrt(-19) where x and y are rational integers. Plotted as points on a plane, they can be joined in a grid of isosceles triangles or be seen as the center points of hexagonal regions. Adjusting the regions to be regular hexagons makes for appealing diagrams, which we will come to shortly. %C A346723 (To be precise, we map each element, z, to the region of the complex plane containing the points that have z as their nearest ring element, then map these (hexagonal) regions continuously to the cells of a (regular) hexagonal grid.) %C A346723 R is one of 9 related rings that are unique factorization domains, meaning their elements factorize into prime elements in a unique way, just as with rational integers and prime numbers. See the Wikipedia link or the Stark reference, for example. %C A346723 This set of sequences is inspired by tilings: see the Wichmann link. Each tiling represents one of the 9 rings and shows the primes as distinctively colored squares or hexagons as appropriate. %C A346723 6 other rings (of the 9) can be mapped to the hexagonal grid in the same way. See the comments entitled "General properties of the related hexagonal spiral sequences" in A346721. %D A346723 L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910. %D A346723 H. M. Stark, An Introduction to Number Theory. Markham, Chicago, 1970; Theorem 8.22 on page 295 lists the nine UFDs of the form Q(sqrt(-d)), cf. A003173. %H A346723 OEIS Wiki, <a href="http://oeis.org/wiki/Algebraic_integers">Algebraic integers</a>. %H A346723 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ComplexPlane.html">Complex Plane</a>, <a href="https://mathworld.wolfram.com/HexagonalGrid.html">Hexagonal Grid</a>, <a href="https://mathworld.wolfram.com/RingofIntegers.html">Ring of Integers</a>. %H A346723 Brian Wichmann, <a href="http://www.tilingsearch.org/special/ufd.pdf">Tiling for Unique Factorization Domains</a>, Jul 22 2019. See Figure 7. %H A346723 Wikipedia, <a href="https://en.wikipedia.org/wiki/Quadratic_integer">Quadratic integer</a>. %H A346723 Wikipedia, <a href="https://en.wikipedia.org/wiki/Unique_factorization_domain">Unique factorization domain</a>. %F A346723 m is a term if and only if A345764(m) is a term. %e A346723 The sequence is constructed in the same way as A346721, but the relevant prime is 19 instead of 7. See the example section of A346721. %Y A346723 Cf. A003173, A345764. %Y A346723 Norms of primes in R: A341787. %Y A346723 Equivalent sequences for other Q(sqrt(D)): A345436 (D=-1), A345437 (D=-2), A345435 (D=-3), A346721 (D=-7), A346722 (D=-11), A346724 (D=-43), A346725 (D=-67), A346726 (D=-163). %K A346723 nonn %O A346723 1,2 %A A346723 _Peter Munn_, Aug 13 2021