PentatopeNumbers

The pentatope numbers $1, 5, 15, 35, …$ — $\binom{n+3}{4}$ — the 4-simplex figurate numbers.

OEISA000332↗

PentatopeNumbers$\binom{n+3}{4}$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(PentatopeNumbers) = +\infty$, and $At(PentatopeNumbers, k) = \binom{k+3}{4}$ unranks in closed form -- $At(PentatopeNumbers, 5) = 70$.
• OEIS A000332.
• Membership goes through Element by bisecting the monotone quartic for its exact root -- $Element(70, PentatopeNumbers)$ is true, $Element(71, PentatopeNumbers)$ is false.
• $PentatopeNumbers(n) = \sum_{i=1}^n TetrahedralNumbers(i)$, one dimension up from TetrahedralNumbers.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Binomial(_r + 4, 4)

rank(_x)

Apply(
       (sq_at: integer) => sq_at if Binomial(sq_at + 4, 4) == _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if Binomial(sq_at + 2 ^ sq_b + 4, 4) <=
                                          _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 + 4, 4) == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if Binomial(sq_at + 2 ^ sq_b + 4, 4) <=
                                          _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: TetrahedralNumbers, Binomial, Count, At, Element