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 A345764 #23 Aug 05 2021 00:15:41
%S A345764 0,1,2,2,1,2,2,7,8,7,10,11,10,7,8,7,10,11,10,19,20,21,20,19,24,25,25,
%T A345764 24,19,20,21,20,19,24,25,25,24,37,38,39,40,39,38,37,44,45,46,45,44,37,
%U A345764 38,39,40,39,38,37,44,45,46,45,44,61,62,63,64,65,64,63,62,61
%N A345764 Number the tiles of a regular hexagonal tiling from 0 in a spiral. Consider perpendicular axes, X and Y, through the center of tile 0, one of which passes through the center of tile 1. Define a set of equivalence classes of tiles with respect to reflections about X and Y. a(n) is the smallest number of a tile in the same equivalence class as tile n.
%C A345764 The sense of the spiral (clockwise/counterclockwise) and its orientation are not significant, but for the purpose of illustration, we depict a counterclockwise spiral with its first step towards the right side of the page.
%C A345764 Equivalence classes contain a maximum of 4 tiles. This happens when tile m's reflection about axis X is a different tile, m_x, and these 2 tiles' reflections about axis Y are 2 further tiles, m_y and m_xy, to give an equivalence class {m, m_x, m_y, m_xy}. Some equivalence classes are smaller, because a tile is its own reflection about an axis, X or Y, that passes through the center of the tile.
%C A345764 The Wichmann reference describes bijections from certain unique factorization domains to the hexagonal tiling. Align the spiral with the mapping so that domain identities 0 and 1 map to tiles 0 and 1 respectively. If two integers from one of the domains map to tiles in the same equivalence class, then they share the same status as units, primes or composites.
%H A345764 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalGrid.html">Hexagonal Grid</a>.
%H A345764 Brian Wichmann, <a href="http://www.tilingsearch.org/special/ufd.pdf">Tiling for Unique Factorization Domains</a>, Jul 22 2019.
%F A345764 a(n) = min({m : |A307012(m)| = |A307012(n)| and |A328818(m)| = |A328818(n)|}).
%F A345764 a(n) = min({m : |A307012(n)| = |A307012(m)| and |2*A307013(n) - A307012(n)| = |2*A307013(m) - A307012(m)|}).
%e A345764 Illustration of the relative positions of tiles on the spiral, marking the n-th tile on the spiral by a(n) to denote its equivalence class:
%e A345764 .
%e A345764 . 24 -- 25 -- 25 -- 24
%e A345764 . / \
%e A345764 . / \
%e A345764 . 19 10 -- 11 -- 10 19
%e A345764 . / / \ \
%e A345764 . / / \ \
%e A345764 . 20 7 2 --- 2 7 20
%e A345764 . / / / \ \ \
%e A345764 . / / / \ \ \
%e A345764 . 21 8 1 0 --- 1 8 21
%e A345764 . \ \ \ / /
%e A345764 . \ \ \ / /
%e A345764 . 20 7 2 --- 2 --- 7 20
%e A345764 . \ \ /
%e A345764 . \ \ /
%e A345764 . 19 10 -- 11 -- 10 -- 19
%e A345764 . \
%e A345764 . \
%e A345764 . 24 -- 25 -- 25 -- 24
%e A345764 .
%e A345764 Recall that the underlying tile numbers count steps along the spiral from 0. When we follow the spiral in the illustration above and encounter a number m, which denotes an equivalence class, for the first time, this is also at tile number m.
%e A345764 Tile 1 maps to itself (as does tile 4) when reflected about the axis through the centers of tiles 0 and 1 (horizontal as shown above). Tiles 1 and 4 map to each other when reflected about the perpendicular (vertical) axis. So tiles 1 and 4 form an equivalence class, and the smallest number of a tile in this class is 1. So a(1) = 1 and a(4) = 1.
%Y A345764 Cf. A274820, A307012, A307013, A328818.
%K A345764 nonn
%O A345764 0,3
%A A345764 _Peter Munn_, Jun 26 2021