PolygonalNumbers

The $k$-gonal figurate numbers $P(k, n) = \big((k-2)n^2 - (k-4)n\big)/2$, one lazy indexed collection per polygon size $k \ge 3$.

PolygonalNumbers(k)$P(k, n)$ for $n = 1, 2, 3, …$, the $k$-gonal numbers.

Typed signatures

Type	Package	Applies
(integer<0..>) -> indexed_collection<integer>	combinatorics	always

Domain: Collections

Details

• A lazy indexed family for each $k \ge 3$: $Count(PolygonalNumbers(k)) = +\infty$, and $At(PolygonalNumbers(k), n)$ unranks $P(k, n)$ in closed form -- $At(PolygonalNumbers(5), 3) = 12$.
• $k = 3, 4, 5, 6, 7, 8$ reproduce TriangularNumbers, SquareNumbers, PentagonalNumbers, HexagonalNumbers, HeptagonalNumbers and OctagonalNumbers termwise; those fixed-$k$ names exist as their own lazy collections for convenience, not as a separate definition.
• Membership goes through Element by inverting the quadratic in $n$ exactly for the given $k$ -- $Element(12, PolygonalNumbers(5))$ is true, $Element(13, PolygonalNumbers(5))$ is false.

Definition

Epsil, as the implementation runs it.

count(_k)

0 if _k < 3 else PositiveInfinity

unrank(_k, _r)

(_r + 1) * ((_k - 2) * (_r + 1) - (_k - 4)) / 2

rank(_k, _x)

Apply(
       (sq_at: integer) => sq_at if (sq_at + 1) * ((_k - 2) * (sq_at + 1) - (_k -
                                                                             4)) /
                                    2 == _x else -1,
       Fold(
             Function(
                       sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) *
                                           ((_k - 2) * (sq_at + 2 ^ sq_b + 1) -
                                            (_k - 4)) / 2 <= _x else sq_at,
                       sq_at,
                       sq_b,
                     ),
             0,
             Range(Floor(Log(_x, 2)) + 1, 0, -1),
           ),
     ) if _k >= 3 else -1

valid(_k, _x)

Apply(
       (sq_at: integer) => (sq_at + 1) * ((_k - 2) * (sq_at + 1) - (_k - 4)) / 2 ==
                           _x,
       Fold(
             Function(
                       sq_at + 2 ^ sq_b if (sq_at + 2 ^ sq_b + 1) *
                                           ((_k - 2) * (sq_at + 2 ^ sq_b + 1) -
                                            (_k - 4)) / 2 <= _x else sq_at,
                       sq_at,
                       sq_b,
                     ),
             0,
             Range(Floor(Log(_x, 2)) + 1, 0, -1),
           ),
     ) if _k >= 3 && _x >= 1 else False

Enumeration

Examples

7 more kept as data, checked against the oracles

About

Examples	7

Implementation

mappedwolframpackages/oracle/src/mappings.ts

Wolfram's own PolygonalNumber[r, n] gives one term (kernel-verified: matches ours termwise) but there's no lazy collection to Take from -- only Table[PolygonalNumber[r, n], {n, 1, count}], which a per-head template on PolygonalNumbers alone can't compose with the outer Take call. Wolfram Function Repository searched; PolygonalDiagram found but it's a visualization (draws the dot array), not a term source.

See also: TriangularNumbers, SquareNumbers, PentagonalNumbers, HexagonalNumbers, Count, At, Element