A230655 -id:A230655 - cp's OEIS frontend

A357303 Records in the numbers of representations of k^2 as x^2 - xy + y^2, x > 2y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.

1, 2, 3, 5, 8, 14, 23, 41, 68, 122, 203, 365, 608, 851, 1094, 1823, 2552, 3281, 5468, 7655, 9842, 16403, 22964, 29525, 49208, 68891, 82013, 88574, 114818, 147623, 206672, 246038, 265721

Offset: 1

Hugo Pfoertner, Sep 25 2022

The total number of possible representations of A357302(n)^2 = x^2 + x*y + y^2 with integers x,y is 12*a(n) - 6.

			See A357302.

Cf. A002324, A003136, A004016, A230655, A357302.
A246360 seems to be a subsequence.
Cf. A087503.

records[n_] := Module[{ri = n, m = 0, rcs = {}, len}, len = Length[ri];
   While[len > 0, If[First[ri] > m, m = First[ri]; AppendTo[rcs, m]];
    ri = Rest[ri]; len--]; rcs]; (* after Harvey P. Dale in A336679 *)
records[Table[
  Length[FindInstance[
    x^2 - x*y + y^2 == k^2 && x > 2*y && y >= 0, {x, y}, Integers,
    1000]], {k, 0, 2000}]]

A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

Original entry on oeis.org

1, 1, 4, 5, 5, 5, 11, 11, 12, 12, 12, 15, 21, 21, 21, 22, 22, 22, 28, 28, 34, 34, 34, 34, 35, 35, 38, 44, 44, 44, 50, 50, 50, 50, 50, 51, 57, 57, 63, 63, 63, 63, 69, 69, 69, 69, 69, 72, 79, 79, 79, 85, 85, 85, 85, 85, 91, 91, 91, 91, 97, 97, 103, 104, 104, 104

Offset: 1

Hugo Pfoertner, Mar 25 2019

nonn

			a(1)=a(2)=(2/2)=1: 2 solutions (-1,1),(1,-1);
a(3)=(8/2)=4: 2 solutions for n<3 and the 6 solutions (1,1), (-1,-1), (1,-2), (2,-1), (-1,2), (-2,1);
a(4)=a(5)=a(6)=10/2=5: 8 solutions for n<6 and the 2 solutions (-2,2), (2,-2).

Cf. A014201, A014202, A198799, A230655, A300419.

cp's OEIS Frontend

A357303 Records in the numbers of representations of k^2 as x^2 - xy + y^2, x > 2y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

Views

Author

Keywords

Examples

Crossrefs

cp's OEIS Frontend

A357303 Records in the numbers of representations of k^2 as x^2 - x*y + y^2, x > 2*y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A307109 Number of integer solutions to x^2 + x*y + y^2 <= n excluding (0,0), divided by 2.

Views

Author

Keywords

Examples

Crossrefs

A357303 Records in the numbers of representations of k^2 as x^2 - xy + y^2, x > 2y >= 0, corresponding to the numbers of grid points with squared radius A357302(n)^2 in an angular sector 0 <= phi < Pi/6 of the triangular lattice.