PentagonalNumbers

The pentagonal numbers $1, 5, 12, 22, …$ — $P(n) = n(3n-1)/2$ — dots in a pentagon.

OEISA000326↗Wolfram LanguagePolygonalNumber↗SageMathsage.combinat.combinat.polygonal_number(5, n)↗

PentagonalNumbers$P(n) = n(3n-1)/2$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(PentagonalNumbers) = +\infty$, and $At(PentagonalNumbers, k) = k(3k-1)/2$ unranks in closed form -- $At(PentagonalNumbers, 5) = 35$.
• OEIS A000326.
• Membership goes through Element by inverting the closed form exactly, solving the quadratic in $n$ over the integers -- $Element(35, PentagonalNumbers)$ is true, $Element(36, PentagonalNumbers)$ is false.
• The $k$-gonal case $k=5$ of PolygonalNumbers; not to be confused with the (signed-index) generalized pentagonal numbers of Euler's pentagonal number theorem.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

(_r + 1) * (3 * (_r + 1) - 1) / 2

rank(_x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) * (3 * (sq_at + 1) - 1) / 2 ==
                                    _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (3 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    1) / 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) * (3 * (sq_at + 1) - 1) / 2 == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (3 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    1) / 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, HexagonalNumbers, Count, At, Element