A383466 -id:A383466 - cp's OEIS frontend

1, 8, 38, 92, 170, 272, 398, 548, 722, 920, 1142, 1388, 1658, 1952, 2270, 2612, 2978, 3368, 3782, 4220, 4682, 5168, 5678, 6212, 6770, 7352, 7958, 8588, 9242, 9920, 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

Offset: 0

Scott R. Shannon and N. J. A. Sloane, Jul 22 2025

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.
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.
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.
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.
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.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Scott R. Shannon, Illustration for a(1) = 8 [Note that the cell counts shown on these five figures do not include the black exterior region, so the totals are off by 1]
Scott R. Shannon, Illustration for a(2) = 38
Scott R. Shannon, Illustration for a(3) = 92
Scott R. Shannon, Illustration for a(8) = 722
Scott R. Shannon, Illustration for a(10) = 1142
N. J. A. Sloane, Sketch to illustrate a(2) = 38. The two hexagrams are colored red and black, respectively.
N. J. A. Sloane, Sketch to illustrate a(3) = 92. The three hexagrams are colored red, blue, and black, respectively.
N. J. A. Sloane, Analogous illustration for three pentagrams
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A094159, A152746.

See A077588, A069894, and A383466 for analogous sequences based on triangles, squares, and pentagrams.

A386477[n_] := If[n == 0, 1, 6*n*(2*n - 1) + 2]; Array[A386477, 50, 0] (* or *)
Join[{1}, 6*PolygonalNumber[6, Range[49]] + 2] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 8, 38, 92}, 50] (* Paolo Xausa, Jul 24 2025 *)

From Stefano Spezia, Jul 23 2025: (Start)
G.f.: (1 + 5*x + 17*x^2 + x^3)/(1 - x)^3.
E.g.f.: 2*exp(x)*(1 + 3*x + 6*x^2) - 1. (End)
a(n) = A152746(n) + 2, for n >= 1. - Paolo Xausa, Jul 24 2025

cp's OEIS Frontend

A386477 a(0) = 1; thereafter a(n) = 2(6n^2 - 3*n + 1).

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