HexagonalNumbers

The hexagonal numbers $1, 6, 15, 28, …$ — $H(n) = n(2n-1)$ — dots in a hexagon.

OEISA000384↗Wolfram LanguagePolygonalNumber↗SageMathsage.combinat.combinat.polygonal_number(6, n)↗

HexagonalNumbers$H(n) = n(2n-1)$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(HexagonalNumbers) = +\infty$, and $At(HexagonalNumbers, k) = k(2k-1)$ unranks in closed form -- $At(HexagonalNumbers, 5) = 45$.
• OEIS A000384.
• Membership goes through Element by inverting the closed form exactly -- $Element(45, HexagonalNumbers)$ is true, $Element(44, HexagonalNumbers)$ is false.
• Every hexagonal number is triangular ($H(n) = T(2n-1)$); the $k$-gonal case $k=6$ of PolygonalNumbers.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

(_r + 1) * (4 * (_r + 1) - 2) / 2

rank(_x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) * (4 * (sq_at + 1) - 2) / 2 ==
                                    _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (4 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    2) / 2 <= _x else sq_at,
                      sq_at,
                      sq_b,
                    ), 0, Range(Floor(Log(_x, 2)) + 1, 0, -1)),
     ) if True else -1

valid(_x)

Apply(
       (sq_at: integer) => (sq_at + 1) * (4 * (sq_at + 1) - 2) / 2 == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (4 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    2) / 2 <= _x else sq_at,
                      sq_at,
                      sq_b,
                    ), 0, Range(Floor(Log(_x, 2)) + 1, 0, -1)),
     ) if True && _x >= 1 else False

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEISWolfram LanguageSageMath

See also: PolygonalNumbers, TriangularNumbers, PentagonalNumbers, Count, At, Element