A105584 - OEIS

%I #13 Oct 01 2016 21:17:57

%S 1,2,1,4,1,2,3,2,1,2,1,4,3,4,1,4,1,2,1,4,1,2,3,2,3,4,3,2,1,2,3,2,1,2,

%T 1,4,1,2,3,2,1,2,1,4,3,4,1,4,3,4,3,2,3,4,1,4,1,2,1,4,3,4,1,4,1,2,1,4,

%U 1,2,3,2,1,2,1,4,3,4,1,4,1,2,1,4,1,2,3,2,3,4,3,2,1,2,3,2,3,4,3,2,3,4,1,4,3

%N Fixed point of the morphism 1 -> 34, 2 -> 32, 3 -> 12, 4 -> 14, starting from a(0) = 1.

%C A triangle space fill substitution: characteristic polynomial:x^4-2*x^3-2*x^2-4*x.

%C This triangle set was obtained by shifting the Heighway's dragon matrix about: M(Heighways's)={{1, 1, 0, 0}, {0, 1, 1, 0}, {0, 0, 1, 1}, {1, 0, 0, 1}} M(triangle)={{0, 0, 1, 1}, {0, 1, 1, 0}, {1, 1, 0, 0}, {1, 0, 0, 1}} This result is a permutation of the rows of the matrix. I have obtained three triangle sets and two Heighway's sets by experiments like these.

%H F. M. Dekking, <a href="http://dx.doi.org/10.1016/0001-8708(82)90066-4">Recurrent Sets</a>, Advances in Mathematics, vol. 44, no.1, April 1982, page 96, section 4.11.

%H <a href="/index/Fi#FIXEDPOINTS">Index entries for sequences that are fixed points of mappings</a>

%t Flatten[ Nest[ Flatten[ # /. {1 -> {3, 4}, 2 -> {3, 2}, 3 -> {1, 2}, 4 -> {1, 4}} &], {1}, 8]] (* _Robert G. Wilson v_, May 07 2005 *)

%K nonn

%O 0,2

%A _Roger L. Bagula_, May 03 2005

%E Edited by _Robert G. Wilson v_, May 07 2005