CenteredHexagonalNumbers

The centered hexagonal numbers $1, 7, 19, 37, …$ — $3n^2-3n+1$ — a hexagon of dots grown ring by ring around a center.

OEISA003215↗

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

Domain: Collections

Details

• A lazy indexed collection: $Count(CenteredHexagonalNumbers) = +\infty$, and $At(CenteredHexagonalNumbers, k) = 3k^2-3k+1$ unranks in closed form -- $At(CenteredHexagonalNumbers, 5) = 61$.
• OEIS A003215.
• Membership goes through Element by inverting the closed form exactly -- $Element(61, CenteredHexagonalNumbers)$ is true, $Element(60, CenteredHexagonalNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

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

rank(_x)

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