A386477 -id:A386477 - cp's OEIS frontend

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

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

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).

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).

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

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

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 *)

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)

cp's OEIS Frontend

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

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