StarNumbers

The star numbers $1, 13, 37, 73, …$ — $6n^2-6n+1$ — centered figurate numbers of a six-pointed star.

OEISA003154↗

StarNumbers$6n^2-6n+1$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(StarNumbers) = +\infty$, and $At(StarNumbers, k) = 6k^2-6k+1$ unranks in closed form -- $At(StarNumbers, 5) = 121$.
• OEIS A003154.
• Membership goes through Element by inverting the closed form exactly -- $Element(121, StarNumbers)$ is true, $Element(120, StarNumbers)$ is false.
• Every star number is both centered hexagonal and centered triangular in disguise (it is the centered figurate number of the $\{6/2\}$ hexagram).

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

6 * (_r + 1) * (_r + 1 - 1) + 1

rank(_x)

Apply(
       (sq_at: integer) => sq_at if 6 * (sq_at + 1) * (sq_at + 1 - 1) + 1 == _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if 6 * (sq_at + 2 ^ sq_b + 1) *
                                          (sq_at + 2 ^ sq_b + 1 - 1) + 1 <= _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) => 6 * (sq_at + 1) * (sq_at + 1 - 1) + 1 == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if 6 * (sq_at + 2 ^ sq_b + 1) *
                                          (sq_at + 2 ^ sq_b + 1 - 1) + 1 <= _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	OEIS

See also: CenteredHexagonalNumbers, Count, At, Element