A339623 Consider a square drawn on the perimeter of a square lattice with side length n. a(n) is the number of regions inside the square after drawing unit circles centered at each interior lattice point of the square.
1, 5, 21, 52, 97, 156, 229, 316, 417, 532, 661, 804, 961, 1132, 1317, 1516, 1729, 1956, 2197, 2452, 2721, 3004, 3301, 3612, 3937, 4276, 4629, 4996, 5377, 5772, 6181, 6604, 7041, 7492, 7957, 8436, 8929, 9436, 9957, 10492, 11041, 11604, 12181, 12772, 13377, 13996, 14629, 15276, 15937, 16612, 17301
Offset: 1
Links
- Index entries for linear recurrences with constant coefficients, signature (3,-3,1).
- Peter Kagey, Example of a(4) = 52.
Programs
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Magma
[1,5] cat [7*n^2-18*n+12 : n in [3..80]];
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Mathematica
Join[{1, 5}, LinearRecurrence[{3, -3, 1}, {21, 52, 97}, 49]] (* Amiram Eldar, Dec 10 2020 *)
Formula
a(n) = 7*n^2 - 18*n + 12 for n >= 3, with a(1) = 1, a(2) = 5.
a(n) = A186862(n)/8+1 for n >= 3. - Hugo Pfoertner, Dec 10 2020
From Stefano Spezia, Dec 10 2020: (Start)
G.f.: x*(1 + 2*x + 9*x^2 + 3*x^3 - x^4)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)