THE WYTHOFF ARRAY an array of Fibonacci-like rows that contains every positive integer exactly once
Split the positive integers into two interleaved streams by the golden ratio: the lower⌊nφ⌋ = 1,3,4,6,8,… and the upper⌊nφ²⌋ = 2,5,7,10,… — two Beatty sequences that are disjoint and cover everything (every integer in exactly one). Stack the Fibonacci-like rows built from these and you get the Wythoff array: every positive integer appears once, and every Fibonacci-style sequence is a row of it. It is the losing positions of Wythoff's game (take from one pile, or equally from both) — the golden ratio as a partition of the integers.
THE TECHNIQUE Beatty split ⌊nφ⌋ / ⌊nφ²⌋: disjoint, covering; b(n)−a(n)=n
The demo shows the two golden Beatty streams interleaving to cover every integer once, with the identity b(n)−a(n)=n: live demo
HISTORY & CREDIT Wythoff 1907 · Beatty 1926
“You can't split the integers evenly by an irrational.” — φ does exactly that: two Beatty sequences, disjoint, covering all — the arithmetic of Wythoff's game. cited
1907 · Willem Abraham Wythoff — the game and its golden-ratio losing positions. 1926 · Samuel Beatty — the theorem: ⌊nα⌋, ⌊nβ⌋ partition the integers iff 1/α+1/β=1 (here α=φ, β=φ²). the array · Morrison / Kimberling — every integer once, every Fibonacci row present.
An irrational that partitions the whole numbers cleanly — and an array holding each of them exactly once. Wythoff / Beatty
RECOMMEND FOR I-13 the Beatty identity, on the compiler
On the canonical compiler, for the golden Beatty split, b(3) − a(3) = 7 − 4 = 3 = n — the identity that pairs the two streams:
$ i13 run g_wythoff.i13 # the golden Beatty split: b(n) - a(n) = n
RUN OK · 26 step(s) · peak stack 5 · call depth 0
diff = 3 -- b(3) - a(3) = 7 - 4
equals_n = 1 -- b(n) - a(n) = n : the streams are golden-paired
Recommend: the Wythoff array is φ as a partition of the integers — two golden Beatty streams that split the whole numbers with no overlap and no gap, holding each exactly once. i13 grounds the pairing b(n)−a(n)=n. Not a keeper (a partition is a structure the ratio induces, not a mechanism a correct alternative lacks), but the dart that shows φ organizes the integers themselves, and the game theory hiding in the golden ratio.