SquarePyramidalNumbers

The square pyramidal numbers $1, 5, 14, 30, …$ — $n(n+1)(2n+1)/6$ — stacked squares, partial sums of the square numbers.

OEISA000330↗

SquarePyramidalNumbers$n(n+1)(2n+1)/6$ for $n = 1, 2, 3, …$.

Domain: Collections

Details

• A lazy indexed collection: $Count(SquarePyramidalNumbers) = +\infty$, and $At(SquarePyramidalNumbers, k) = k(k+1)(2k+1)/6$ unranks in closed form -- $At(SquarePyramidalNumbers, 5) = 55$.
• OEIS A000330.
• Membership goes through Element by bisecting the monotone cubic for its exact root -- $Element(55, SquarePyramidalNumbers)$ is true, $Element(56, SquarePyramidalNumbers)$ is false.
• $SquarePyramidalNumbers(n) = \sum_{i=1}^n SquareNumbers(i)$.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Binomial(_r + 3, 3) + Binomial(_r + 2, 3)

rank(_x)

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