A261849 Number of squares in an n X n grid that are enclosed in a circle of diameter n (having the same center as the grid).

0, 0, 1, 4, 9, 16, 21, 32, 45, 60, 69, 88, 101, 120, 145, 164, 185, 216, 241, 276, 293, 332, 365, 392, 437, 476, 509, 556, 593, 648, 681, 732, 785, 832, 885, 936, 989, 1052, 1109, 1176, 1225, 1288, 1353, 1428, 1489, 1560, 1625, 1696, 1781, 1860, 1933, 2016, 2085, 2180, 2241, 2340, 2425, 2512, 2609, 2700, 2793, 2876, 2973, 3080, 3173

Offset: 1

Author

V.J. Pohjola, Sep 03 2015

Keywords

• nonn
• easy

Comments

a(1)=0 by definition.

The idea behind the sequence was originally proposed at http://www.sanaristikot.net on Aug 19 2015 by Jaska Himberg.

Links

• Yi Yang, Table of n, a(n) for n = 1..10000
• V. J. Pohjola ("Olavi Kivalo") and Jaakko Himberg ("Jaska"), 8545.Lukujono16
• Giovanni Resta, Representation of a(3)-a(11)

Crossrefs

Cf. A085690, A242118.

Programs

Mathematica

c[n_, i_, j_] := Ceiling[Sqrt[(n - 2 i)^2 + (n - 2 j)^2]];
t1[q_] := Take[q, 1]; t2[p_] := Take[p, -1]; p2[r_] := Power[r, 2];
area = {}; (Do[
  a = {}; (Do[
    If[c[n, i, j] == n || c[n, i, j] == n - 1 || c[n, i, j] == n - 2,
     AppendTo[a, {i, j}]], {i, 1, Ceiling[n/2 (1 - Sqrt[2]/2)]}, {j, i,
     Floor[n/2]}]);
  b = (n - 2*Map[t2, Flatten[Map[t1, GatherBy[a, First]], 1]]);
  sum1 = 4*Apply[Plus, Drop[b, -1]]; sum2 = Map[p2, Last[b]];
  AppendTo[area, (sum1 + sum2)], {n, 2, 100}]);
Flatten[{0, area}]
a[1] = 0; a[n_] := If[EvenQ[n], 4 Sum[ Floor[ Sqrt[(n/2)^2 - k^2]], {k, n/2}], 4 Floor[n/2] - 3 + 4 Sum[Floor[-1/2 + Sqrt[(n/2)^2 - (k + 1/2)^2]], {k, n/2 - 1}]]; Array[a, 60] (* Giovanni Resta, Sep 10 2015 *)