A sequence with |xn| = xn-1 + xn+1 has period 9

Recently I heard the theorem that any (in both directions) infinite sequence of real numbers xn such that for all n

|xn| = xn-1 + xn+1


has a period of length 9. Here is my proof.

p + r ( 0)
r ( 0)
-p ( 0)
p - r ( 0)
2p - r ( 0)
p ( 0)
r - p ( 0)
- r ( 0)
p ( 0)
p + r
From (0) we conclude (i) that the sequence contains a nonnegative element, (ii) that one of its neighbours is nonnegative, and (iii) that at least one of the two elements adjacent to a pair of nonnegative neighbours is nonnegative. More precisely: the sequence contains in some direction a triple of adjacent elements of the form (p, p+r, r) with 0 r p. To the left we have extended the sequence with another 8 elements. From (0) we further conclude that the whole sequence is determined by a pair of adjacent values; hence, the repetition of the pair (p, p+r) at distance 9 proves the theorem. [The above deserves recording for its lack of case analyses.]

Plataanstraat 5
The Netherlands
1 September 1983
prof.dr. Edsger W. Dijkstra
Burroughs Research Fellow

Transcriber: Kevin Hely.
Last revised on Tue, 24 Jun 2003.