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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.

%I A339623 #20 Sep 08 2022 08:46:26
%S A339623 1,5,21,52,97,156,229,316,417,532,661,804,961,1132,1317,1516,1729,
%T A339623 1956,2197,2452,2721,3004,3301,3612,3937,4276,4629,4996,5377,5772,
%U A339623 6181,6604,7041,7492,7957,8436,8929,9436,9957,10492,11041,11604,12181,12772,13377,13996,14629,15276,15937,16612,17301
%N 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.
%H A339623 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).
%H A339623 Peter Kagey, <a href="/A339623/a339623_1.png">Example of a(4) = 52</a>.
%F A339623 a(n) = 7*n^2 - 18*n + 12 for n >= 3, with a(1) = 1, a(2) = 5.
%F A339623 a(n) = A186862(n)/8+1 for n >= 3. - _Hugo Pfoertner_, Dec 10 2020
%F A339623 From _Stefano Spezia_, Dec 10 2020: (Start)
%F A339623 G.f.: x*(1 + 2*x + 9*x^2 + 3*x^3 - x^4)/(1 - x)^3.
%F A339623 a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n >= 3. (End)
%t A339623 Join[{1, 5}, LinearRecurrence[{3, -3, 1}, {21, 52, 97}, 49]] (* _Amiram Eldar_, Dec 10 2020 *)
%o A339623 (Magma) [1,5] cat [7*n^2-18*n+12 : n in [3..80]];
%Y A339623 Cf. A186862, A339609 (triangular version).
%K A339623 nonn,easy
%O A339623 1,2
%A A339623 _Wesley Ivan Hurt_, Dec 10 2020