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 A386477 #60 Jul 24 2025 10:52:26
%S A386477 1,8,38,92,170,272,398,548,722,920,1142,1388,1658,1952,2270,2612,2978,
%T A386477 3368,3782,4220,4682,5168,5678,6212,6770,7352,7958,8588,9242,9920,
%U A386477 10622,11348,12098,12872,13670,14492,15338,16208,17102,18020,18962,19928,20918,21932,22970,24032,25118,26228,27362,28520,29702,30908,32138,33392,34670
%N A386477 a(0) = 1; thereafter a(n) = 2*(6*n^2 - 3*n + 1).
%C A386477 Definition: A regular hexagram of radius R is formed by placing six equally-spaced points P_0 .. P_5 around the boundary of a circle of radius R, and drawing line segments P_0 - P_2 - P_4 - P_0 and P_1 - P_3 - P_5 - P_1.
%C A386477 Theorem 1: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular hexagrams with the same radius and the same center.
%C A386477 Conjecture 2: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular hexagrams with any radii and any centers.
%C A386477 The following construction works for any n >= 1. Take 6*n equally-spaced points P_i around a circle, and draw hexagrams starting at P_i for i = 0, ..., n-1.
%C A386477 The resulting planar graph decomposes into 6*n triangular cells each with 2*n-1 cells (see the red triangle in the "Three pentagons" illustration), plus the interior and exterior regions, for a total of 12*n^2 - 6*n + 2 regions. There are 12*n^2 vertices, for n>0.
%H A386477 Paolo Xausa, <a href="/A386477/b386477.txt">Table of n, a(n) for n = 0..10000</a>
%H A386477 Scott R. Shannon, <a href="/A386477/a386477.png">Illustration for a(1) = 8</a> [Note that the cell counts shown on these five figures do not include the black exterior region, so the totals are off by 1]
%H A386477 Scott R. Shannon, <a href="/A386477/a386477_1.png">Illustration for a(2) = 38</a>
%H A386477 Scott R. Shannon, <a href="/A386477/a386477_2.png">Illustration for a(3) = 92</a>
%H A386477 Scott R. Shannon, <a href="/A386477/a386477_3.png">Illustration for a(8) = 722</a>
%H A386477 Scott R. Shannon, <a href="/A386477/a386477_4.png">Illustration for a(10) = 1142</a>
%H A386477 N. J. A. Sloane, <a href="/A386477/a386477.jpg">Sketch to illustrate a(2) = 38. The two hexagrams are colored red and black, respectively.</a>
%H A386477 N. J. A. Sloane, <a href="/A386477/a386477_1.jpg">Sketch to illustrate a(3) = 92. The three hexagrams are colored red, blue, and black, respectively.</a>
%H A386477 N. J. A. Sloane, <a href="/A383466/a383466.jpg">Analogous illustration for three pentagrams</a>
%H A386477 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A386477 From _Stefano Spezia_, Jul 23 2025: (Start)
%F A386477 G.f.: (1 + 5*x + 17*x^2 + x^3)/(1 - x)^3.
%F A386477 E.g.f.: 2*exp(x)*(1 + 3*x + 6*x^2) - 1. (End)
%F A386477 a(n) = A152746(n) + 2, for n >= 1. - _Paolo Xausa_, Jul 24 2025
%t A386477 A386477[n_] := If[n == 0, 1, 6*n*(2*n - 1) + 2]; Array[A386477, 50, 0] (* or *)
%t A386477 Join[{1}, 6*PolygonalNumber[6, Range[49]] + 2] (* or *)
%t A386477 LinearRecurrence[{3, -3, 1}, {1, 8, 38, 92}, 50] (* _Paolo Xausa_, Jul 24 2025 *)
%Y A386477 Cf. A094159, A152746.
%Y A386477 See A077588, A069894, and A383466 for analogous sequences based on triangles, squares, and pentagrams.
%K A386477 nonn,easy
%O A386477 0,2
%A A386477 _Scott R. Shannon_ and _N. J. A. Sloane_, Jul 22 2025