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 A329515 #28 Sep 08 2022 08:46:24 %S A329515 1,1,2,1,3,2,1,3,6,2,1,3,6,7,2,1,3,6,9,6,2,1,3,6,9,12,6,2,1,3,6,9,12, %T A329515 12,6,2,1,3,6,9,12,15,10,6,2,1,3,6,9,12,15,18,10,6,2,1,3,6,9,12,15,18, %U A329515 17,10,6,2,1,3,6,9,12,15,18,21,14,10,6,2 %N A329515 Array read by upward antidiagonals: row n = coordination sequence for cylinder formed by rolling up a strip of width 2*n+1 hexagons cut from the hexagonal grid by cuts perpendicular to grid lines. %C A329515 The width of the strip is a little harder to define here. In the illustration for n=2, the strip is five 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+2 edges (10 edges in the illustration for n=2). %C A329515 Because the width 2*n+1 is odd, one edge of the cut has to be displaced sideways (by half the width of a hexagon) in order for the two edges to mesh properly. The arrows in the figures indicate pairs of points which will coalesce. %C A329515 For the case when the strip is 2*n hexagons wide see A329512. %C A329515 For the case when the cuts are parallel to the grid lines, see A329508. %C A329515 See A329501 and A329504 for coordination sequences for cylinders formed by rolling up the square grid. %H A329515 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 A329515 N. J. A. Sloane, <a href="/A329515/a329515.pdf">Illustration for row n = 0</a>, showing vertices of cylinder of width (or circumference) 1 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce. %H A329515 N. J. A. Sloane, <a href="/A329515/a329515_1.pdf">Illustration for row n = 1</a>, showing vertices of cylinder of width (or circumference) 3 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce. %H A329515 N. J. A. Sloane, <a href="/A329515/a329515_2.pdf">Illustration for row n = 2</a>, showing vertices of cylinder of width (or circumference) 5 labeled with distance from base point 0. The cylinder is formed by identifying the black lines. Arrows indicate two points which will coalesce. %H A329515 <a href="/index/Con#coordination_sequences">Index entries for coordination sequences</a> %F A329515 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 A329515 The g.f. G(n) for row n (n>=0) is (strongly) conjectured to be %F A329515 (1/(1-x))*(1 + 2*x + 3*x^2*(1-x^(2*n-1))/(1-x) - (n-2)*x^(2*n+1) - n*x^(2*n+2)). %F A329515 The values of G(0) through G(6) (certified by MAGMA) are: %F A329515 (1 + x)/(1 - x), %F A329515 (x^4 - x^3 - 3*x^2 - 2*x - 1)/(x - 1), %F A329515 (2*x^6 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1), %F A329515 (3*x^8 + x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1), %F A329515 (4*x^10 + 2*x^9 - 3*x^8 - 3*x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1), %F A329515 (5*x^12 + 3*x^11 - 3*x^10 - 3*x^9 - 3*x^8 - 3*x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1), %F A329515 (6*x^14 + 4*x^13 - 3*x^12 - 3*x^11 - 3*x^10 - 3*x^9 - 3*x^8 - 3*x^7 - 3*x^6 - 3*x^5 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1). %F A329515 Note that row n is equal to 4*n+2 once the (2*n+2)-nd term has been reached. %F A329515 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 A329515 Array begins: %e A329515 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, ... %e A329515 1, 3, 6, 7, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, ... %e A329515 1, 3, 6, 9, 12, 12, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 17, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 22, 18, 18, 18, 18, 18, 18, 18, 18, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 27, 22, 22, 22, 22, 22, 22, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 32, 26, 26, 26, 26, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 37, 30, 30, 30, 30, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 42, 34, 34, 34, 34, 34, 34, 34, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 47, 38, 38, 38, 38, 38, ... %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 52, 42, 42, 42, ... %e A329515 ... %e A329515 The initial antidiagonals are: %e A329515 1, %e A329515 1, 2, %e A329515 1, 3, 2, %e A329515 1, 3, 6, 2, %e A329515 1, 3, 6, 7, 2, %e A329515 1, 3, 6, 9, 6, 2, %e A329515 1, 3, 6, 9, 12, 6, 2, %e A329515 1, 3, 6, 9, 12, 12, 6, 2, %e A329515 1, 3, 6, 9, 12, 15, 10, 6, 2, %e A329515 1, 3, 6, 9, 12, 15, 18, 10, 6, 2, %e A329515 1, 3, 6, 9, 12, 15, 18, 17, 10, 6, 2, %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 14, 10, 6, 2, %e A329515 1, 3, 6, 9, 12, 15, 18, 21, 24, 14, 10, 6, 2, %e A329515 ... %o A329515 (Magma) %o A329515 n := 2; \\ set n %o A329515 R<x> := RationalFunctionField(Integers()); %o A329515 FG3<R, S, T> := FreeGroup(3); %o A329515 Q3 := quo<FG3| R^2, S^2, T^2, R*S*T = T*S*R, (S*T*S*R)^n*S*R >; %o A329515 H := AutomaticGroup(Q3); %o A329515 f3 := GrowthFunction(H); %o A329515 PSR := PowerSeriesRing(Integers():Precision := 60); %o A329515 Coefficients(PSR!f3); %o A329515 // 1, 3, 6, 9, 12, 12, 10, 10, 10, 10, 10, 10, ... (row n) %o A329515 f3; // G(n) %o A329515 // (2*x^6 - 3*x^4 - 3*x^3 - 3*x^2 - 2*x - 1)/(x - 1) %Y A329515 Rows 0,1,2,3 are A040000, A329516, A329517, A329771. %Y A329515 Cf. A008486, A329501-A329514. %K A329515 nonn,tabl,easy %O A329515 0,3 %A A329515 _N. J. A. Sloane_, Nov 25 2019