TriangularNumbers

The triangular numbers $1, 3, 6, 10, …$ — $T(n) = n(n+1)/2$ — dots in a triangle.

OEISA000217↗Wolfram LanguagePolygonalNumber↗SageMathsage.combinat.combinat.polygonal_number(3, n)↗

TriangularNumbers$T(n) = n(n+1)/2$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(TriangularNumbers) = +\infty$, and $At(TriangularNumbers, k) = k(k+1)/2$ unranks in closed form -- $At(TriangularNumbers, 5) = 15$.
• OEIS A000217.
• Membership goes through Element by inverting the closed form exactly: $x$ is triangular iff $8x+1$ is a perfect square -- $Element(15, TriangularNumbers)$ is true, $Element(14, TriangularNumbers)$ is false.
• The $k$-gonal case $k=3$ of PolygonalNumbers.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

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

rank(_x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) * (1 * (sq_at + 1) - -1) / 2 ==
                                    _x else -1,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (1 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    -1) / 2 <= _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) * (1 * (sq_at + 1) - -1) / 2 == _x,
       Fold(Function(
                      sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) * (1 * (sq_at +
                                                                         2 ^
                                                                         sq_b + 1) -
                                                                    -1) / 2 <= _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

Properties #

live ↗

In$15∈\mathrm{TriangularNumbers}$Out$\mathrm{True}$

live ↗

In$14∈\mathrm{TriangularNumbers}$Out$\mathrm{False}$

About

Examples	3
Crosswalk	OEISWolfram LanguageSageMath

See also: PolygonalNumbers, SquareNumbers, PentagonalNumbers, Count, At, Element