CenteredTriangularNumbers

The centered triangular numbers $1, 4, 10, 19, …$ — $(3n^2-3n+2)/2$ — a triangle of dots grown ring by ring around a center.

OEISA005448↗

CenteredTriangularNumbers$(3n^2-3n+2)/2$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(CenteredTriangularNumbers) = +\infty$, and $At(CenteredTriangularNumbers, k) = (3k^2-3k+2)/2$ unranks in closed form -- $At(CenteredTriangularNumbers, 5) = 31$.
• OEIS A005448.
• Membership goes through Element by inverting the closed form exactly -- $Element(31, CenteredTriangularNumbers)$ is true, $Element(30, CenteredTriangularNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

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

rank(_x)

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