|
%I #18 Sep 11 2019 01:20:22
%S 1,0,0,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,
%T 1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,
%U 1,0,1,0,0,0,1,0,1,0,0,0,1,0,1,0,1,0
%N Fixed point of the morphism 0 -> 10, 1 -> 1000.
%C Prefixing 0 gives A284751.
%H Clark Kimberling, <a href="/A285301/b285301.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>
%F Conjecture: a(n) = A284751(n+1). - _R. J. Mathar_, May 08 2017
%F From _Michel Dekking_, Sep 11 2019: (Start)
%F Proof of Mathar's conjecture.
%F Let sigma be the morphism 0 -> 10, 1 -> 1000.
%F Let tau be the morphism 0 -> 01, 1 -> 0001.
%F Then A284751 is the fixed point of tau. So it suffices to prove that
%F 0 sigma^n(1) = tau^n(0) 0 for all n>0.
%F This formula follows by induction, using that tau and sigma are conjugate morphisms: 1 tau(w) = sigma(w) 1 for all words w.
%F (Plug in w = tau^n(0) in tau^{n+1}(0)).
%F (End)
%e 0 -> 10-> 100010 -> 1000101010100010 ->
%t s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0, 0}}] &, {0}, 10]; (* A285301 *)
%t Flatten[Position[s, 0]]; (* A285302 *)
%t Flatten[Position[s, 1]]; (* A086398 *)
%Y Cf. A284302, A086398, A284751.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Apr 25 2017
|
