login
a(n) gives the number of squares in the n-th iteration of the Harter-Heighway dragon.
0

%I #39 Jun 15 2024 15:25:42

%S 0,0,0,1,4,11,30,78,205,546,1455,4062,11192,31889,88487,254594,710683,

%T 2047705,5711439,16455169,45894868,132118562,368149344,1059171430,

%U 2950384277,8484556353

%N a(n) gives the number of squares in the n-th iteration of the Harter-Heighway dragon.

%C a(n) >= A003230(n-4) for n >= 4, and a(n) > A003230(n-4) for n >= 7.

%H Code Golf Stack Exchange user Domenico Modica, <a href="https://codegolf.stackexchange.com/q/216919/53884">Eye test - How many squares are in this picture?</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Dragon_curve#Heighway_dragon">Dragon Curve</a>

%e For n = 5, the fifth iteration of the Harter-Heighway dragon has a(5) = 4 squares, as illustrated below. All of the squares are 1 X 1.

%e *---*

%e | |

%e *---* *

%e |

%e *---*

%e |

%e *---*---* *---* *---*

%e | | | | | | |

%e *---* *---*---*---* *---*

%e | | |

%e *---*---* *---*

%e | |

%e *---*

%e For n = 7, the seventh iteration of the Harter-Heighway dragon has a(7) = 30 squares, as illustrated below. It contains A003230(7) = 28 1 X 1 squares and two 2 X 2 squares.

%e _ _

%e |_|_| |_

%e _ _| _|

%e |_|_|_

%e |_|_|

%e |_ _ _ _

%e _ _|_|_| _|_| _|_|

%e |_|_|_|_|_ _|_|_ _|_|_ _

%e |_| |_| |_|_|_|_|_| |_| |_|_|

%e |_|_|_|_ |_

%e _ _|_| |_| _|

%e |_|_|_|_ |_|

%e |_| |_|

%Y A003230 gives the number of 1 X 1 squares.

%K nonn,more

%O 1,5

%A _Peter Kagey_, Jan 05 2021

%E a(17)-a(26) from _Bert Dobbelaere_, Jun 15 2024