PellNumbers

The Pell numbers $0, 1, 2, 5, 12, 29, …$ as a lazy indexed collection.

OEISA000129↗

PellNumbers$P_n = 2P_{n-1} + P_{n-2}$, $P_0 = 0$, $P_1 = 1$, an infinite indexed collection.

Domain: Collections

Details

• A lazy indexed collection: $Count(PellNumbers) = +\infty$, and $At(PellNumbers, 1) = P_0 = 0$.
• OEIS A000129, starting exactly at its offset-0 term: $0, 1, 2, 5, 12, 29, 70, …$; strictly increasing from the start.
• Membership goes through Element: $Element(12, PellNumbers)$ is true, $Element(7, PellNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Fold(
      Function([wk_w[2], 2 * wk_w[2] + wk_w[1]], wk_w, wk_n),
      [0, 1],
      Range(0, _r - 1, 1),
    )[1]

rank(_x)

Fold(
      Function(
                wk_w if wk_n >= 0 && wk_w[1] > _x else [
                                                         wk_w[2],
                                                         2 * wk_w[2] + wk_w[1],
                                                         wk_n if wk_w[3] < 0 &&
                                                                 wk_w[1] == _x else wk_w[3],
                                                       ],
                wk_w,
                wk_n,
              ),
      [0, 1, -1],
      Range(
             0,
             2 * (Apply(
                         (fl_r: list<integer>) => fl_r[2],
                         Fold(Function(
                                        [
                                          (fl_s[1] - fl_s[1] % 2 ^ fl_k) / 2 ^
                                                                           fl_k,
                                          fl_s[2] + fl_k,
                                        ] if fl_s[1] >= 2 ^ fl_k else fl_s,
                                        fl_s,
                                        fl_k,
                                      ), [_x + 2, 0], [
                                                       8_192,
                                                       4_096,
                                                       2_048,
                                                       1_024,
                                                       512,
                                                       256,
                                                       128,
                                                       64,
                                                       32,
                                                       16,
                                                       8,
                                                       4,
                                                       2,
                                                       1,
                                                     ]),
                       ) + 1) + 12,
             1,
           ),
    )[3] if _x >= 0 else -1

valid(_x)

Fold(
      Function(
                wk_w if wk_n >= 0 && wk_w[1] > _x else [
                                                         wk_w[2],
                                                         2 * wk_w[2] + wk_w[1],
                                                         wk_n if wk_w[3] < 0 &&
                                                                 wk_w[1] == _x else wk_w[3],
                                                       ],
                wk_w,
                wk_n,
              ),
      [0, 1, -1],
      Range(
             0,
             2 * (Apply(
                         (fl_r: list<integer>) => fl_r[2],
                         Fold(Function(
                                        [
                                          (fl_s[1] - fl_s[1] % 2 ^ fl_k) / 2 ^
                                                                           fl_k,
                                          fl_s[2] + fl_k,
                                        ] if fl_s[1] >= 2 ^ fl_k else fl_s,
                                        fl_s,
                                        fl_k,
                                      ), [_x + 2, 0], [
                                                       8_192,
                                                       4_096,
                                                       2_048,
                                                       1_024,
                                                       512,
                                                       256,
                                                       128,
                                                       64,
                                                       32,
                                                       16,
                                                       8,
                                                       4,
                                                       2,
                                                       1,
                                                     ]),
                       ) + 1) + 12,
             1,
           ),
    )[3] >= 0 if _x >= 0 else False

Enumeration

Examples

1 more kept as data, checked against the oracles

About

Examples	1
Crosswalk	OEIS

See also: Count, At, Element, JacobsthalNumbers, CentralDelannoyNumbers