A364352 - OEIS

A364352

a(n) is the number of regions into which the plane is divided by n lines parallel to each edge of an equilateral triangle with side n such that the lines extend the parallel edge and divide the other edges into unit segments.

7, 16, 30, 49, 73, 102, 136, 175, 219, 268, 322, 381, 445, 514, 588, 667, 751, 840, 934, 1033, 1137, 1246, 1360, 1479, 1603, 1732, 1866, 2005, 2149, 2298, 2452, 2611, 2775, 2944, 3118, 3297, 3481, 3670, 3864, 4063, 4267, 4476, 4690, 4909, 5133, 5362, 5596, 5835, 6079, 6328

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OFFSET

1,1

COMMENTS

Detailed instructions for drawing the lines. Along the edges of an equilateral triangle with side n, points are marked that divide the edges into unit segments. Draw all infinite straight lines that connect those points and are parallel to the edges of the triangle. For n = 1..5, the link shows the construction of these lines.

LINKS

Paolo Xausa, Table of n, a(n) for n = 1..10000

Nicolay Avilov, Illustration of initial terms

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

a(n) = n*(5*n + 3)/2 + 3;

a(n) = A147875(n) + 3 = A134238(n+1) + 2.

From Stefano Spezia, Nov 23 2023: (Start)

O.g.f.: x*(7 - 5*x + 3*x^2)/(1 - x)^3.

E.g.f.: exp(x)*(3 + 4*x + 5*x^2/2) - 3. (End)

EXAMPLE

a(1) = 1 + 3 + 3 = 7;