Invariant Sequences

Substitution Invariant Beatty Sequences

with θ irrational and taken to satisfy 0 < θ < 1; plainly this may be assumed without loss of generality. Evidently (fn) is a sequence of zeros and ones. Denote by w0 and w1 words on the alphabet {0, 1} ; that is, finite strings in the letters 0 and 1. Then the sequence (fn) is said to be invariant under the substitution W given by W : 0 −→ w0, 1 −→ w1, if the infinite strings fθ = f1f2f3 . . ....

Substitution invariant cutting sequences

Invariant Sequences under Binomial Transformation

The classical binomial inversion formula states that an = Hk=o(k)(~ty^k ( ~ ®> ̂ 2,...) if and only if hn = Tl=Q(l)(-l)ak (n = 0,1,2,...). In this paper we study those sequences {an} such that Hk=0(l)(-lfak = ±an (n = 0,1? 2,...). If EJU(Z)H)*"* = <**("* °), we say that {aj is an invariant sequence. If Efc=0(*)(~l)** = -an (n>0)9 we say that {an} is an inverse invariant sequence. Throughout this...

Invariant games and non-homogeneous Beatty sequences

We characterize all the pairs of complementary non-homogenous Beatty sequences (An) n≥0 and (Bn) n≥0 for which there exists an invariant game having exactly {(An, Bn) | n ≥ 0} ∪ {(Bn, An) | n ≥ 0} as set of P-positions. Using the notion of Sturmian word and tools arising in symbolic dynamics and combinatorics on words, this characterization can be translated to a decision procedure relying only...

Nonhomogeneous Beatty Sequences Leading to Invariant Games

We characterize pairs of complementary non-homogeneous Beatty sequences (An)n>0 and (Bn)n>0, with the restriction A1 = 1 and B1 ≥ 3, for which there exists an invariant take-away game having {(An, Bn), (Bn, An) | n > 0} ∪ {(0, 0)} as set of P -positions. Using the notion of Sturmian word arising in combinatorics on words, this characterization can be translated into a decision procedure relying...