Generate Beatty sequence terms floor(n*alpha) with complementary sequence verification
Alpha (irrational > 1)
Number of Terms
Result
Beatty Sequence (floor(n*alpha))
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Complementary (floor(n*beta))
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Detailed Derivation
Beatty Sequence Definition
B(alpha) = floor(n*alpha) for n = 1,2,3...
Complementary: 1/alpha + 1/beta = 1
Rayleigh theorem: sequences partition N
alpha = phi (~1.618) is the most famous
Beatty sequences, named after Samuel Beatty, are sequences of the form floor(n*alpha) where alpha is an irrational number greater than 1. The Rayleigh theorem states that two Beatty sequences with complementary alpha and beta partition the positive integers.
⚠Alpha must be irrational and > 1 for the Rayleigh theorem to hold. Rational alpha produces overlapping sequences.
What Is a Beatty Sequence?
A Beatty sequence is a sequence of integers generated by taking the floor of multiples of an irrational number. These sequences have the remarkable property that complementary pairs partition the natural numbers, a result discovered by Lord Rayleigh and later generalized by Beatty.
Golden Ratio Case
alpha = phi = 1.618, beta = phi^2 = 2.618. B(phi): 1,3,4,6,8,9,11... B(phi^2): 2,5,7,10,13,15,18... They partition N.
The Wythoff Nim game uses Beatty sequences of phi. The losing positions are (floor(n*phi), floor(n*phi^2)). This connects game theory to number theory.
History
Rayleigh (1894) discovered the property for sqrt(2). Beatty (1926) generalized it as a problem in the American Mathematical Monthly. Also called homogeneous Beatty sequences.
Game TheoryNumber TheoryCombinatoricsWythoff GameIrrationals StudySequence Analysis
FAQs about Beatty Sequences
What is a Beatty sequence?▼
floor(n*alpha) for irrational alpha > 1. Example alpha=sqrt(2): 1,2,4,5,7,8,9,11,12,14... Named after Samuel Beatty.
What is Rayleigh theorem?▼
If 1/alpha+1/beta=1, then Beatty sequences of alpha and beta partition all positive integers without overlap or gaps.
What is the most famous Beatty sequence?▼
alpha = phi = 1.618 (golden ratio). B(phi)=1,3,4,6,8,9,11... B(phi^2)=2,5,7,10,13,15,18... Used in Wythoff Nim.
Can alpha be rational?▼
No - the Rayleigh theorem requires alpha to be irrational. For rational alpha, the sequences would have overlaps or gaps and would not partition N perfectly.
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