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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.

%I A364352 #41 Nov 24 2023 12:21:56
%S A364352 7,16,30,49,73,102,136,175,219,268,322,381,445,514,588,667,751,840,
%T A364352 934,1033,1137,1246,1360,1479,1603,1732,1866,2005,2149,2298,2452,2611,
%U A364352 2775,2944,3118,3297,3481,3670,3864,4063,4267,4476,4690,4909,5133,5362,5596,5835,6079,6328
%N 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.
%C A364352 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.
%H A364352 Paolo Xausa, <a href="/A364352/b364352.txt">Table of n, a(n) for n = 1..10000</a>
%H A364352 Nicolay Avilov, <a href="/A364352/a364352.jpg">Illustration of initial terms</a>
%H A364352 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%F A364352 a(n) = n*(5*n + 3)/2 + 3;
%F A364352 a(n) = A147875(n) + 3 = A134238(n+1) + 2.
%F A364352 From _Stefano Spezia_, Nov 23 2023: (Start)
%F A364352 O.g.f.: x*(7 - 5*x + 3*x^2)/(1 - x)^3.
%F A364352 E.g.f.: exp(x)*(3 + 4*x + 5*x^2/2) - 3. (End)
%e A364352 a(1) = 1 + 3 + 3 = 7;
%e A364352 a(2) = 2^2 + 3*3 + 3 = 16;
%e A364352 a(5) = 5^2 + 3*9 + 3*6 + 3 = 73.
%t A364352 LinearRecurrence[{3,-3,1},{7,16,30},100] (* _Paolo Xausa_, Oct 16 2023 *)
%Y A364352 Cf. A134238, A147875, A177862, A343755, A364401.
%K A364352 nonn,easy
%O A364352 1,1
%A A364352 _Nicolay Avilov_, Jul 20 2023
%E A364352 Edited by _Peter Munn_, Sep 02 2023