LucasNumbers

The Lucas numbers $2, 1, 3, 4, 7, 11, …$ as a lazy indexed collection.

OEISA000032↗Wolfram LanguageLucasL↗SageMathsage.combinat.combinat.lucas_number2(n, 1, -1)↗

LucasNumbers$L_n = L_{n-1} + L_{n-2}$, $L_0 = 2$, $L_1 = 1$, an infinite indexed collection.

Domain: Collections

Details

• A lazy indexed collection: $Count(LucasNumbers) = +\infty$, and $At(LucasNumbers, 1) = L_0 = 2$.
• OEIS A000032, starting exactly at its offset-0 term: $2, 1, 3, 4, 7, 11, 18, …$. The sequence dips once ($L_0 = 2 > L_1 = 1$) before climbing forever from $L_1$ on.
• Membership goes through Element: $Element(1, LucasNumbers)$ is true, $Element(5, LucasNumbers)$ is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

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

rank(_x)

Fold(
      Function(
                wk_w if wk_n >= 1 && wk_w[1] > _x else [
                                                         wk_w[2],
                                                         wk_w[1] + wk_w[2],
                                                         wk_n if wk_w[3] < 0 &&
                                                                 wk_w[1] == _x else wk_w[3],
                                                       ],
                wk_w,
                wk_n,
              ),
      [2, 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 >= 1 && wk_w[1] > _x else [
                                                         wk_w[2],
                                                         wk_w[1] + wk_w[2],
                                                         wk_n if wk_w[3] < 0 &&
                                                                 wk_w[1] == _x else wk_w[3],
                                                       ],
                wk_w,
                wk_n,
              ),
      [2, 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	OEISWolfram LanguageSageMath

See also: Count, At, Element, FibonacciNumbers