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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.
1
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
1,1
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.
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;
a(2) = 2^2 + 3*3 + 3 = 16;
a(5) = 5^2 + 3*9 + 3*6 + 3 = 73.
MATHEMATICA
LinearRecurrence[{3, -3, 1}, {7, 16, 30}, 100] (* Paolo Xausa, Oct 16 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Nicolay Avilov, Jul 20 2023
EXTENSIONS
Edited by Peter Munn, Sep 02 2023
STATUS
approved