ThueMorseNumbers

The Thue–Morse sequence $0, 1, 1, 0, 1, 0, 0, 1, …$ as a lazy indexed collection.

OEISA010060↗SageMathsage.combinat.words.word_generators.words.ThueMorseWord()↗

ThueMorseNumbers$t(n) = popcount(n) \bmod 2$, an infinite indexed collection.

Domain: Collections

Details

• A lazy indexed collection: $Count(ThueMorseNumbers) = +\infty$, and $At(ThueMorseNumbers, 1) = t(0) = 0$.
• OEIS A010060, starting exactly at its offset-0 term: $0, 1, 1, 0, 1, 0, 0, 1, 1, 0, …$.
• Not monotone; every term is $0$ or $1$, so membership is exactly $\{0, 1\}$ -- $Element(0, ThueMorseNumbers)$ and $Element(1, ThueMorseNumbers)$ are true, everything else (including $2$) is false.

Definition

Epsil, as the implementation runs it.

count()

PositiveInfinity

unrank(_r)

Fold(Function(
               [(bt_s[1] - bt_s[1] % 2) / 2, (bt_s[2] + bt_s[1] % 2) % 2],
               bt_s,
               bt_b,
             ), [_r, 0], Range(1, Floor(Log(_r + 1, 2)) + 2, 1))[2]

rank(_x)

_x if _x == 0 || _x == 1 else -1

valid(_x)

_x == 0 || _x == 1

Enumeration

Examples

1 more kept as data, checked against the oracles

Basic #

The first 20 terms, OEIS A010060

live ↗

In$\mathrm{Take}(\mathrm{ThueMorseNumbers}, 20)$Out$\bigl\lbrack0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1\bigr\rbrack$

About

Examples	2
Crosswalk	OEISSageMath

See also: Count, At, Element, SternDiatomicSequence