A383466 a(0) = 1; thereafter a(n) = 10*n^2 - 5*n + 2.

1, 7, 32, 77, 142, 227, 332, 457, 602, 767, 952, 1157, 1382, 1627, 1892, 2177, 2482, 2807, 3152, 3517, 3902, 4307, 4732, 5177, 5642, 6127, 6632, 7157, 7702, 8267, 8852, 9457, 10082, 10727, 11392, 12077, 12782, 13507, 14252, 15017, 15802, 16607, 17432, 18277, 19142, 20027, 20932, 21857, 22802, 23767, 24752, 25757, 26782, 27827

Offset: 0

Author

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

Keywords

• nonn
• easy
• changed

Comments

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.

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.

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.

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.

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).

Links

• Paolo Xausa, Table of n, a(n) for n = 0..10000
• Scott R. Shannon, Illustration for a(1) = 7. [Note that the cell counts shown on these four figures do not include the black exterior region, so the totals are off by 1]
• Scott R. Shannon, Illustration for a(2) = 32.
• Scott R. Shannon, Illustration for a(3) = 77.
• Scott R. Shannon, Illustration for a(8) = 602.
• N. J. A. Sloane, Illustration for a(1) = 7.
• N. J. A. Sloane, Illustration for a(2) = 32.
• N. J. A. Sloane, Illustration for a(n), n >= 1, showing a(3) = 77.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

See A077588, A069894, and A386477 for analogous sequences based on triangles, squares, and hexagrams.

Without the "+2" in the definition, the sequence is A152745.

Programs

Mathematica

A383466[n_] := If[n == 0, 1, 5*n*(2*n - 1) + 2]; Array[A383466, 50, 0] (* or *)
Join[{1}, 5*PolygonalNumber[6, Range[49]] + 2] (* or *)
LinearRecurrence[{3, -3, 1}, {1, 7, 32, 77}, 50] (* Paolo Xausa, Jul 22 2025 *)

Formula

From Elmo R. Oliveira, Sep 03 2025: (Start)

G.f.: (1 + 4*x + 14*x^2 + x^3)/(1 - x)^3.

E.g.f.: exp(x)*(2 + 5*x + 10*x^2) - 1.

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)