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.
2, 4, 8, 6, 14, 24, 8, 20, 28, 42, 10, 26, 38, 56, 74, 12, 32, 48, 66, 82, 104, 14, 38, 58, 80, 100, 122, 138, 16, 40, 64, 88, 114, 134, 164, 186, 18, 46, 74, 98, 132, 152, 186, 212, 240, 20, 52, 84, 112, 150, 174, 208, 232, 266, 304, 22, 58, 94, 126, 160, 196, 226, 262, 296, 324, 362
Offset: 1
Examples
The triangle T(n,k) begins: n\k 1 2 3 4 5 6 7 8 9 10 11 ... 1: 2 2: 4 8 3: 6 14 24 4: 8 20 28 42 5: 10 26 38 56 74 6: 12 32 48 66 82 104 7: 14 38 58 80 100 122 138 8: 16 40 64 88 114 134 164 186 9: 18 46 74 98 132 152 186 212 240 10: 20 52 84 112 150 174 208 232 266 304 11: 22 58 94 126 160 196 226 262 296 324 362 ... See linked illustration of the term T(5,3) = 38.
Links
- Felix Huber, Rows n = 1..141 of triangle, flattened
- Felix Huber, Illustraton of T(5,3) = 38
- Eric Weisstein's World of Mathematics, Ellipse
- Eric Weisstein's World of Mathematics, Pick's Theorem
Programs
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Maple
T386539:=proc(n,k) local x,y,p,s; p:=2*(n+k); s:={0}; for x to n-1 do y:=floor(k*sqrt(1-x^2/n^2)); p:=p+4*y; s:=s union {y} od; return p-2*nops(s) end proc; seq(seq(T386539(n,k),k=1..n),n=1..25); -
Mathematica
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 *)
Comments