PronicNumbers

The pronic (oblong) numbers $2, 6, 12, 20, …$ — $n(n+1)$ — twice a triangular number.

OEISA002378↗

PronicNumbers$n(n+1)$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(PronicNumbers) = +\infty$, and $At(PronicNumbers, k) = k(k+1)$ unranks in closed form -- $At(PronicNumbers, 5) = 30$.
• OEIS A002378, whose offset differs from ours: A002378 opens $a(0) = 0$; this collection starts at $n=1$, so $At(PronicNumbers, 1) = 2$ and the value $0$ is not a member.
• Membership goes through Element by inverting the closed form exactly -- $Element(30, PronicNumbers)$ is true, $Element(29, PronicNumbers)$ is false. $PronicNumbers(k) = 2 \cdot TriangularNumbers(k)$.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

(_r + 1) * (_r + 1 + 1)

rank(_x)

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