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

Offset: 1

Author

Nicolay Avilov, Jul 20 2023

Keywords

• nonn
• easy

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.

Examples

			a(1) = 1 + 3 + 3 = 7;
a(2) = 2^2 + 3*3 + 3 = 16;
a(5) = 5^2 + 3*9 + 3*6 + 3 = 73.
		

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).

Crossrefs

Cf. A134238, A147875, A177862, A343755, A364401.

Programs

Mathematica

LinearRecurrence[{3,-3,1},{7,16,30},100] (* Paolo Xausa, Oct 16 2023 *)

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)

Extensions

Edited by Peter Munn, Sep 02 2023