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.
0, 0, 4, 10, 22, 39, 61, 88, 120, 157, 199, 246, 298, 355, 417, 484, 556, 633, 715, 802, 894, 991, 1093, 1200, 1312, 1429, 1551, 1678, 1810, 1947, 2089, 2236, 2388, 2545, 2707, 2874, 3046, 3223, 3405, 3592, 3784, 3981, 4183, 4390, 4602, 4819, 5041, 5268, 5500, 5737
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
- Peter Kagey, Example for a(4) = 10.
Programs
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Mathematica
Join[{0, 0, 4}, Table[(5 n^2 - 21 n + 24)/2, {n, 4, 60}]]
Formula
a(n) = (5*n^2 - 21*n + 24)/2 for n >= 4, with a(1)=a(2)=0, a(3)=4.
a(n) = A005476(n-2)+1 for n >= 4. - Hugo Pfoertner, Dec 10 2020
From Stefano Spezia, Dec 10 2020: (Start)
G.f.: x^3*(4 - 2*x + 4*x^2 - x^3)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 4. (End)