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 A383466 #60 Sep 03 2025 09:33:57
%S A383466 1,7,32,77,142,227,332,457,602,767,952,1157,1382,1627,1892,2177,2482,
%T A383466 2807,3152,3517,3902,4307,4732,5177,5642,6127,6632,7157,7702,8267,
%U A383466 8852,9457,10082,10727,11392,12077,12782,13507,14252,15017,15802,16607,17432,18277,19142,20027,20932,21857,22802,23767,24752,25757,26782,27827
%N A383466 a(0) = 1; thereafter a(n) = 10*n^2 - 5*n + 2.
%C A383466 Definition: A regular pentagram of radius R is formed by placing five equally-spaced points P_0 .. P_4 around the boundary of a circle of radius R, and drawing line segments P_0 - P_2 - P_4 - P_1 - P_3 - P_0.
%C A383466 Theorem 1: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular pentagrams with the same radius and the same center.
%C A383466 Conjecture 2: a(n) is the maximum number of regions that can be formed in the plane by drawing n regular pentagrams with any radii and any centers.
%C A383466 The following construction works for any n >= 1. Take 5*n equally-spaced points P_i around a circle, and draw a pentagram through P_i, P_{i+n}, P_{i+2*n}, P_{i+3*n}, P_{i+4*n} for i = 0, ..., n-1.
%C A383466 The resulting planar graph decomposes into 5*n triangular regions each with 2*n-1 cells (see the red triangle in "Illustration for a(n)..."), plus the interior and exterior regions, for a total of 10*n^2 - 5*n + 2 regions. There are 10*n^2 vertices (10 for n=1, 40 for n=2, and so on).
%H A383466 Paolo Xausa, <a href="/A383466/b383466.txt">Table of n, a(n) for n = 0..10000</a>
%H A383466 Scott R. Shannon, <a href="/A383466/a383466_2.png">Illustration for a(1) = 7</a>. [Note that the cell counts shown on these four figures do not include the black exterior region, so the totals are off by 1]
%H A383466 Scott R. Shannon, <a href="/A383466/a383466_3.png">Illustration for a(2) = 32</a>.
%H A383466 Scott R. Shannon, <a href="/A383466/a383466_4.png">Illustration for a(3) = 77</a>.
%H A383466 Scott R. Shannon, <a href="/A383466/a383466_5.png">Illustration for a(8) = 602</a>.
%H A383466 N. J. A. Sloane, <a href="/A383466/a383466.png">Illustration for a(1) = 7</a>.
%H A383466 N. J. A. Sloane, <a href="/A383466/a383466_1.png">Illustration for a(2) = 32</a>.
%H A383466 N. J. A. Sloane, <a href="/A383466/a383466.jpg">Illustration for a(n), n >= 1, showing a(3) = 77</a>.
%H A383466 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A383466 From _Elmo R. Oliveira_, Sep 03 2025: (Start)
%F A383466 G.f.: (1 + 4*x + 14*x^2 + x^3)/(1 - x)^3.
%F A383466 E.g.f.: exp(x)*(2 + 5*x + 10*x^2) - 1.
%F A383466 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%t A383466 A383466[n_] := If[n == 0, 1, 5*n*(2*n - 1) + 2]; Array[A383466, 50, 0] (* or *)
%t A383466 Join[{1}, 5*PolygonalNumber[6, Range[49]] + 2] (* or *)
%t A383466 LinearRecurrence[{3, -3, 1}, {1, 7, 32, 77}, 50] (* _Paolo Xausa_, Jul 22 2025 *)
%Y A383466 See A077588, A069894, and A386477 for analogous sequences based on triangles, squares, and hexagrams.
%Y A383466 Without the "+2" in the definition, the sequence is A152745.
%K A383466 nonn,easy,changed
%O A383466 0,2
%A A383466 _Scott R. Shannon_ and _N. J. A. Sloane_, Jul 22 2025