A105500 - OEIS

%I #28 Jul 13 2024 17:51:35

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

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

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

%N Trajectory of 1 under the morphism 1->{1,2}, 2->{3,2}, 3->{3,4}, 4->{1,4}.

%C Harter-Heighway dragon when interpreting 1, 2, 3, and 4 respectively as unit edge to right, up, left, and down. - _Joerg Arndt_, Jun 03 2021

%C The characteristic polynomial of the transition matrix is x^4-4*x^3+6*x^2-4*x = x*(x-2)*(x^2 - 2*x + 2).

%H Joerg Arndt, <a href="/A105500/a105500.pdf">Iterate six of the Harter-Heighway dragon</a>

%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 (1982), 78-104; page 89, section 4.5.

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

%F a(n) = A246960(n) + 1. - _Joerg Arndt_, Jun 03 2021

%t Flatten[ Nest[ Flatten[ # /. {1 -> {1, 2}, 2 -> {3, 2}, 3 -> {3, 4}, 4 -> {1, 4}} &], {1}, 7]]

%o (Python)

%o def A105500(n): return ((n^(n>>1)).bit_count()&3)+1 # _Chai Wah Wu_, Jul 13 2024

%Y Cf. A246960 (as 0..3).

%Y Indices of terms 1..4: A043724, A043725, A043726, A043727.

%K nonn

%O 0,2

%A _Roger L. Bagula_, May 02 2005