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

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

Author

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

Keywords

• nonn
• easy

Comments

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.

Links

• 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).

Crossrefs

Cf. A094159, A152746.

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

Programs

Mathematica

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 *)

Formula

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