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 A329512 #39 Sep 08 2022 08:46:24 %S A329512 1,1,3,1,3,4,1,3,6,4,1,3,6,9,4,1,3,6,9,9,4,1,3,6,9,12,8,4,1,3,6,9,12, %T A329512 15,8,4,1,3,6,9,12,15,14,8,4,1,3,6,9,12,15,18,12,8,4,1,3,6,9,12,15,18, %U A329512 21,12,8,4 %N A329512 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width 2*n hexagons cut from the hexagonal grid by cuts perpendicular to grid lines. %C A329512 The width of the strip is a little harder to define here. In the illustration for n=2, the strip is four hexagons wide if measured along hexagons that touch edge-to-edge. A path joining two vertices to be identified when the cylinder is formed has length 4n edges (8 edges in the illustration for n=2). %C A329512 For the case when the strip is 2*n+1 hexagons wide see A329515. %C A329512 For the case when the cuts are parallel to the grid lines, see A329508. %C A329512 See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid. %H A329512 Chaim Goodman-Strauss and N. J. A. Sloane, <a href="https://doi.org/10.1107/S2053273318014481">A Coloring Book Approach to Finding Coordination Sequences</a>, Acta Cryst. A75 (2019), 121-134, also <a href="http://NeilSloane.com/doc/Cairo_final.pdf">on NJAS's home page</a>. Also <a href="http://arxiv.org/abs/1803.08530">arXiv:1803.08530</a>. %H A329512 N. J. A. Sloane, <a href="/A329512/a329512_1.pdf">Illustration for row n = 1</a>, showing vertices of cylinder of width (or circumference) 2 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce. %H A329512 N. J. A. Sloane, <a href="/A329512/a329512_2.pdf">Illustration for row n = 2</a>, showing vertices of cylinder of width (or circumference) 4 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce. %H A329512 <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a> %F A329512 The g.f.s for the rows could be found using the "trunks and branches" method (see Goodman-Strauss and Sloane), as was done in A329508. This step has not yet been carried out, so the following g.f. is at present only conjectural. %F A329512 The g.f. G(n) for row n (n>=1) is (strongly) conjectured to be %F A329512 (1/(1-x))*(1 + 2*x + 3*x^2*(1-x^(2*n-2))/(1-x) - (n-2)*x^(2*n) - (n-1)*x^(2*n+1)). %F A329512 The values of G(1) through G(6) (certified by MAGMA) are: %F A329512 (1+x)^2/(1-x), %F A329512 (x^3-2*x^2-1)*(1+x)^2/(x-1), %F A329512 (2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1), %F A329512 (3*x^7-4*x^6+2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1), %F A329512 (4*x^9-5*x^8+3*x^7-4*x^6+2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1), %F A329512 (5*x^11-6*x^10+4*x^9-5*x^8+3*x^7-4*x^6+2*x^5-3*x^4+x^3-2*x^2-1)*(1+x)^2/(x-1). %F A329512 Note that row n is equal to 4*n once the (2*n+1)-st term has been reached. %F A329512 The g.f.s for the rows can also be obtained by regarding the 1-skeleton of the cylinder as the Cayley diagram for an appropriate group H, and computing the growth function for H (see the MAGMA code). %e A329512 Array begins: %e A329512 1, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, ... %e A329512 1, 3, 6, 9, 9, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, ... %e A329512 1, 3, 6, 9, 12, 15, 14, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, ... %e A329512 1, 3, 6, 9, 12, 15, 18, 21, 19, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, ... %e A329512 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 24, 20, 20, 20, 20, 20, 20, 20, 20, 20, ... %e A329512 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 29, 24, 24, 24, 24, 24, 24, 24, ... %e A329512 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 34, 28, 28, 28, 28, 28, ... %e A329512 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 39, 32, 32, 32, ... %e A329512 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 44, 36, 36, ... %e A329512 ... %e A329512 The initial antidiagonals are: %e A329512 1, %e A329512 1, 3, %e A329512 1, 3, 4, %e A329512 1, 3, 6, 4, %e A329512 1, 3, 6, 9, 4, %e A329512 1, 3, 6, 9, 9, 4, %e A329512 1, 3, 6, 9, 12, 8, 4, %e A329512 1, 3, 6, 9, 12, 15, 8, 4, %e A329512 1, 3, 6, 9, 12, 15, 14, 8, 4, %e A329512 1, 3, 6, 9, 12, 15, 18, 12, 8, 4, %e A329512 ... %o A329512 (Magma) %o A329512 n := 2; \\ set n %o A329512 R<x> := RationalFunctionField(Integers()); %o A329512 FG3<R, S, T> := FreeGroup(3); %o A329512 Q3 := quo<FG3| R^2, S^2, T^2, R*S*T = T*S*R, (S*T*S*R)^(2*n) >; %o A329512 H := AutomaticGroup(Q3); %o A329512 f3 := GrowthFunction(H); %o A329512 PSR := PowerSeriesRing(Integers():Precision := 60); %o A329512 Coefficients(PSR!f3); %o A329512 // 1, 3, 6, 9, 12, 15, 14, 12, 12, 12, 12, 12, 12, 12, 12, ... (row n) %o A329512 f3; // G(n) %o A329512 // (x^3-2*x^2-1)*(1+x)^2/(x-1) %Y A329512 Rows 1,2,3 are A113311, A329513, A329514. %Y A329512 Cf. A008486, A329501-A329517. %K A329512 nonn,tabl,easy %O A329512 1,3 %A A329512 _N. J. A. Sloane_, Nov 24 2019