cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A288248 Number of essentially different simple n-sided lattice polygons with minimum area subject to the conditions of A288247.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 1, 3, 2, 10, 6, 5, 11, 4, 11, 19, 17, 55
Offset: 3

Views

Author

Hugo Pfoertner, Jun 07 2017

Keywords

Comments

Polygons differing only by rotation and/or reflection are counted only once.

Links

Crossrefs

Cf. A288247, A288250.

Extensions

Based on an exhaustive search performed by Michael Collier, a(12) corrected by Hugo Pfoertner, Jun 10 2017

A288250 Number of essentially different simple n-sided lattice polygons with maximum area subject to the conditions of A288247.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 2, 3, 5, 1, 15, 5, 8, 2, 1, 8, 2, 7, 1
Offset: 3

Views

Author

Hugo Pfoertner, Jun 07 2017

Keywords

Comments

Polygons differing only by rotation and/or reflection are counted only once.
A conjectured continuation of the sequence starting at a(22) is 6, 3, 3, 6, 19, 36, 61.

References

  • For references and links see A288247.

Links

Crossrefs

Cf. A288247, A288248, A288249.

A288249 2 * largest possible area of a simple n-sided lattice polygon whose vertex coordinates x and y are both independent permutations of the integers 1 ... n, subject to the condition that none of its edges are mutually parallel.

Original entry on oeis.org

3, 8, 18, 28, 44, 65, 88, 118, 151, 189, 226, 272, 323, 377, 433, 493, 558, 627, 701, 778, 860, 945, 1034, 1127, 1224, 1325
Offset: 3

Views

Author

Hugo Pfoertner, Jun 07 2017

Keywords

Comments

It is conjectured that a(n) = 2*n^2 - 9*(n-1) for all n>22.

References

  • For references and links see A288247.

Crossrefs

Cf. A288247, A288250.

A386538 a(n) is the maximum possible area of a polygon within a circle of radius n, where both the center and the vertices lie on points of a unit square grid.

Original entry on oeis.org

0, 2, 8, 24, 42, 74, 104, 138, 186, 240, 304, 362, 424, 512, 594, 690, 776, 880, 986, 1104, 1232, 1346, 1490, 1624, 1762, 1930, 2088, 2256, 2418, 2594, 2784, 2962, 3170, 3368, 3584, 3810, 4008, 4248, 4466, 4730, 4976, 5210, 5474, 5736, 6024, 6306, 6570, 6864, 7154
Offset: 0

Views

Author

Felix Huber, Aug 05 2025

Keywords

Comments

a(n) > 99% of the circle area for n >= 50.
Conjecture: The maximum possible area of a polygon within the circle would be the same if only the vertices but not the center were fixed on grid points.
All terms are even.

Examples

			See linked illustration of the term a(4) = 42.

Links

Crossrefs

Cf. A000217, A066643, A123690, A125228, A288247, A322106, A322107, A357577, A386539.

Programs

  • Maple
    A386538:=proc(n)
    local x,y,p,s;
    p:=4*n;
    s:={};
    for x to n do
        y:=floor(sqrt(n^2-x^2));
        p:=p+4*y;
        s:=s union {y}
    od;
    return p-2*nops(s)
end proc;
seq(A386538(n),n=0..48);
  • Mathematica
    a[n_] := Module[{p=4n},s = {}; Do[ y = Floor[Sqrt[n^2 - x^2]];p = p + 4*y;s = Union[s, {y}],{x,n} ];p - 2*Length[s]];Array[a,49,0] (* James C. McMahon, Aug 19 2025 *)

Formula

a(n) = A386539(A000217(n)) = A386539(n,n) for n >= 1.
a(n) <= A066643(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.

Original entry on oeis.org

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

Views

Author

Felix Huber, Aug 05 2025

Keywords

Comments

The axes of the ellipse are assumed to be aligned with the coordinate axes.
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.
All terms are even.

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

Crossrefs

Cf. A000217, A108126, A288247, A386539.

Programs

  • 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 *)

Formula

a(A000217(n)) = T(n,n) = A386538(n).
a(n) < Pi*n*k.
Showing 1-5 of 5 results.

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