OctagonalNumbers

The octagonal numbers $1, 8, 21, 40, …$ — $n(3n-2)$ — dots in an octagon.

OEISA000567↗Wolfram LanguagePolygonalNumber↗SageMathsage.combinat.combinat.polygonal_number(8, n)↗

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

Domain: Collections

Details

• A lazy indexed collection: $Count(OctagonalNumbers) = +\infty$, and $At(OctagonalNumbers, k) = k(3k-2)$ unranks in closed form -- $At(OctagonalNumbers, 5) = 65$.
• OEIS A000567.
• Membership goes through Element by inverting the closed form exactly -- $Element(65, OctagonalNumbers)$ is true, $Element(64, OctagonalNumbers)$ is false.
• The $k$-gonal case $k=8$ of PolygonalNumbers.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

(_r + 1) * (6 * (_r + 1) - 4) / 2

rank(_x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) * (6 * (sq_at + 1) - 4) / 2 ==
                                    _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (6 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    4) / 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) * (6 * (sq_at + 1) - 4) / 2 == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (6 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    4) / 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, HeptagonalNumbers, Count, At, Element