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A386539 Triangle read by rows: T(n,k) is the maximum possible area of a polygon within a ellipse with integer axis n and k, where n >= k >= 1 and both the center and the vertices lie on points of a unit square grid.

%I A386539 #19 Aug 19 2025 11:30:26
%S A386539 2,4,8,6,14,24,8,20,28,42,10,26,38,56,74,12,32,48,66,82,104,14,38,58,
%T A386539 80,100,122,138,16,40,64,88,114,134,164,186,18,46,74,98,132,152,186,
%U A386539 212,240,20,52,84,112,150,174,208,232,266,304,22,58,94,126,160,196,226,262,296,324,362
%N A386539 Triangle read by rows: T(n,k) is the maximum possible area of a polygon within a ellipse with integer axis n and k, where n >= k >= 1 and both the center and the vertices lie on points of a unit square grid.
%C A386539 The axes of the ellipse are assumed to be aligned with the coordinate axes.
%C A386539 Conjecture: The maximum possible area of a polygon within the ellipse would be the same if only the vertices but not the center were fixed on grid points.
%C A386539 All terms are even.
%H A386539 Felix Huber, <a href="/A386539/b386539.txt">Rows n = 1..141 of triangle, flattened</a>
%H A386539 Felix Huber, <a href="/A386539/a386539.pdf">Illustraton of T(5,3) = 38</a>
%H A386539 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Ellipse.html">Ellipse</a>
%H A386539 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PicksTheorem.html">Pick's Theorem</a>
%F A386539 a(A000217(n)) = T(n,n) = A386538(n).
%F A386539 a(n) < Pi*n*k.
%e A386539 The triangle T(n,k) begins:
%e A386539    n\k  1   2   3   4   5   6   7   8   9  10  11 ...
%e A386539    1:   2
%e A386539    2:   4   8
%e A386539    3:   6  14  24
%e A386539    4:   8  20  28  42
%e A386539    5:  10  26  38  56  74
%e A386539    6:  12  32  48  66  82 104
%e A386539    7:  14  38  58  80 100 122 138
%e A386539    8:  16  40  64  88 114 134 164 186
%e A386539    9:  18  46  74  98 132 152 186 212 240
%e A386539   10:  20  52  84 112 150 174 208 232 266 304
%e A386539   11:  22  58  94 126 160 196 226 262 296 324 362
%e A386539  ...
%e A386539 See linked illustration of the term T(5,3) = 38.
%p A386539 T386539:=proc(n,k)
%p A386539     local x,y,p,s;
%p A386539     p:=2*(n+k);
%p A386539     s:={0};
%p A386539     for x to n-1 do
%p A386539         y:=floor(k*sqrt(1-x^2/n^2));
%p A386539         p:=p+4*y;
%p A386539         s:=s union {y}
%p A386539     od;
%p A386539     return p-2*nops(s)
%p A386539 end proc;
%p A386539 seq(seq(T386539(n,k),k=1..n),n=1..25);
%t A386539 T[n_, k_] := Module[{p=2*(n+k)},s = {0};Do[ y = Floor[k*Sqrt[1 - x^2/n^2]];p = p + 4*y;s = Union[s, {y}],{x,n-1}];p - 2*Length[s]];Flatten[Table[T[n, k], {n, 1, 11}, {k, 1, n}]] (* _James C. McMahon_, Aug 19 2025 *)
%Y A386539 Cf. A000217, A108126, A288247, A386539.
%K A386539 nonn,tabl
%O A386539 1,1
%A A386539 _Felix Huber_, Aug 05 2025