A339739 a(n) gives the number of squares in the n-th iteration of the Harter-Heighway dragon.

0, 0, 0, 1, 4, 11, 30, 78, 205, 546, 1455, 4062, 11192, 31889, 88487, 254594, 710683, 2047705, 5711439, 16455169, 45894868, 132118562, 368149344, 1059171430, 2950384277, 8484556353

Offset

1,5

Comments

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

Links

Table of n, a(n) for n=1..26.

Code Golf Stack Exchange user Domenico Modica, Eye test - How many squares are in this picture?

Wikipedia, Dragon Curve

Example

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.
    *---*
    |   |
*---*   *
|
*---*
    |
*---*---*   *---*   *---*
|   |   |   |   |   |   |
*---*   *---*---*---*   *---*
            |   |           |
        *---*---*       *---*
        |   |
        *---*
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.
       _   _
      |_|_| |_
   _   _|    _|
  |_|_|_
    |_|_|
      |_   _       _       _
   _   _|_|_|    _|_|    _|_|
  |_|_|_|_|_   _|_|_   _|_|_   _
    |_| |_| |_|_|_|_|_| |_| |_|_|
              |_|_|_|_        |_
           _   _|_| |_|        _|
          |_|_|_|_          |_|
            |_| |_|

Crossrefs

A003230 gives the number of 1 X 1 squares.

Sequence in context: A102281 A269083 A026583 * A291238 A110034 A340824

Adjacent sequences: A339736 A339737 A339738 * A339740 A339741 A339742

Keyword

nonn,more

Author

Peter Kagey, Jan 05 2021

Extensions

a(17)-a(26) from Bert Dobbelaere, Jun 15 2024

Status

approved