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

1,3

Links

Table of n, a(n) for n=1..50.

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Peter Kagey, Example for a(4) = 10.

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)

Mathematica

Join[{0, 0, 4}, Table[(5 n^2 - 21 n + 24)/2, {n, 4, 60}]]

Crossrefs

Cf. A005476, A339623 (square version).

Sequence in context: A341734 A008255 A008031 * A008248 A301243 A177736

Adjacent sequences: A339606 A339607 A339608 * A339610 A339611 A339612

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Dec 09 2020

Status

approved