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A339609 Consider a triangle drawn on the perimeter of a triangular lattice with side length n. a(n) is the number of regions inside the triangle after drawing unit circles centered at each lattice point inside the triangle.

%I A339609 #23 Dec 21 2020 17:10:13
%S A339609 0,0,4,10,22,39,61,88,120,157,199,246,298,355,417,484,556,633,715,802,
%T A339609 894,991,1093,1200,1312,1429,1551,1678,1810,1947,2089,2236,2388,2545,
%U A339609 2707,2874,3046,3223,3405,3592,3784,3981,4183,4390,4602,4819,5041,5268,5500,5737
%N A339609 Consider a triangle drawn on the perimeter of a triangular lattice with side length n. a(n) is the number of regions inside the triangle after drawing unit circles centered at each lattice point inside the triangle.
%H A339609 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H A339609 Peter Kagey, <a href="/A339609/a339609.png">Example for a(4) = 10</a>.
%F A339609 a(n) = (5*n^2 - 21*n + 24)/2 for n >= 4, with a(1)=a(2)=0, a(3)=4.
%F A339609 a(n) = A005476(n-2)+1 for n >= 4. - _Hugo Pfoertner_, Dec 10 2020
%F A339609 From _Stefano Spezia_, Dec 10 2020: (Start)
%F A339609 G.f.: x^3*(4 - 2*x + 4*x^2 - x^3)/(1 - x)^3.
%F A339609 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)
%t A339609 Join[{0, 0, 4}, Table[(5 n^2 - 21 n + 24)/2, {n, 4, 60}]]
%Y A339609 Cf. A005476, A339623 (square version).
%K A339609 nonn,easy
%O A339609 1,3
%A A339609 _Wesley Ivan Hurt_, Dec 09 2020