Is this number Loeschian?

A positive integer $$\k\$$ is a Loeschian number if

• $$\k\$$ can be expressed as $$\i^2 + j^2 + i\times j\$$ for $$\i\$$, $$\j\$$ integers.

For example, the first positive Loeschian numbers are: $$\1\$$ ($$\i=1, j=0\$$); $$\3\$$ ($$\i=j=1\$$); $$\4\$$ ($$\i=2, j=0\$$); $$\7\$$ ($$\i=2, j=1\$$); $$\9\$$ ($$\i=-3, j=3\$$); ... Note that $$\i, j\$$ for a given $$\k\$$ are not unique. For example, $$\9\$$ can also be generated with $$\i=3, j=0\$$.

Other equivalent characterizations of these numbers are:

• $$\k\$$ can be expressed as $$\i^2 + j^2 + i\times j\$$ for $$\i, j\$$ non-negative integers. (For each pair of integers $$\i, j\$$ there's a pair of non-negative integers that gives the same $$\k\$$)

• There is a set of $$\k\$$ contiguous hexagons that forms a tesselation on a hexagonal grid so that corresponding cells are the same distance apart (see illustrations for $$\k = 4\$$ and for $$\k = 7\$$). (Because of this property, these numbers find application in mobile cellular communication networks.)

• See more characterizations in the OEIS page of the sequence.

The challenge

Given a positive integer, output a truthy result if it is a Loeschian number, or a falsy result otherwise.

The program or function should handle (say in less than a minute) inputs up to $$\1000\$$, or up to data type limitations.

Code golf. Shortest wins.

Test cases

The following numbers should output a truthy result:

1, 4, 7, 12, 13, 108, 109, 192, 516, 999

The following numbers should output a falsy result:

2, 5, 10, 42, 101, 102, 128, 150, 501, 1000
• Related (as noted by @PeterTaylor) Aug 4 '16 at 15:35
• note for the brute force algorithm: if you iterate to √k you reduce algorithm complexity from O(n²) to O(n), at expense of some bytes c;
– Rod
Aug 4 '16 at 17:19
• @Titus Oh now I see. For each pair of integers i, j there's a non-negative pair that gives the same k Jan 6 '17 at 13:13
• @uhoh Your actual question seems much more complicated though. Sorry that I cannot contribute to it Jun 5 '20 at 12:59
• Find all integer pairs that produce a given Loeschian number My first question here, I've borrowed from your question quite a bit, but I think the continuity is helpful, and you did such a nice job writing this up.
– uhoh
Oct 11 '20 at 12:43

€¹m§+Πṁ□π2ṡ

Try it online!

Jelly, 9 8 bytes

ŻpḋÄ$€i Try it online! Returns a non-zero integer for Loeschian numbers and $$\0\$$ for non-Loeschian numbers -1 byte thanks to Sisyphus May be a little cheeky to post, as the only reason this is shorter than Leaky's and Dennis’ Jelly answer is due to newer features, but it's different enough from the existing Jelly answers. How it works ŻpḋÄ$€i - Main link. Takes k on the left
Ż        - Yield [0, 1, 2, ..., k]
`      - Use this as both left and right argument for:
p       -   Cartesian power

Try it online!