TetrahedralNumbers

The tetrahedral numbers $1, 4, 10, 20, …$ — $\binom{n+2}{3}$ — stacked triangles, partial sums of the triangular numbers.

OEISA000292↗

TetrahedralNumbers$\binom{n+2}{3}$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(TetrahedralNumbers) = +\infty$, and $At(TetrahedralNumbers, k) = \binom{k+2}{3}$ unranks in closed form -- $At(TetrahedralNumbers, 5) = 35$.
• OEIS A000292.
• Membership goes through Element by bisecting the monotone cubic for its exact root -- $Element(35, TetrahedralNumbers)$ is true, $Element(36, TetrahedralNumbers)$ is false.
• $TetrahedralNumbers(n) = \sum_{i=1}^n TriangularNumbers(i)$; the 4-simplex case is PentatopeNumbers.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Binomial(_r + 3, 3)

rank(_x)

Apply(
       (sq_at: integer) => sq_at if Binomial(sq_at + 3, 3) == _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if Binomial(sq_at + 2 ^ sq_b + 3, 3) <=
                                          _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) => Binomial(sq_at + 3, 3) == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if Binomial(sq_at + 2 ^ sq_b + 3, 3) <=
                                          _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, PentatopeNumbers, Binomial, Count, At, Element