--description--

The quadtree encoding allows us to describe a $2^N×2^N$ black and white image as a sequence of bits (0 and 1). Those sequences are to be read from left to right like this:

• the first bit deals with the complete $2^N×2^N$ region;
• "0" denotes a split:
  • the current $2^n×2^n$ region is divided into 4 sub-regions of dimension $2^{n - 1}×2^{n - 1}$,
  • the next bits contains the description of the top left, top right, bottom left and bottom right sub-regions - in that order;
• "10" indicates that the current region contains only black pixels;
• "11" indicates that the current region contains only white pixels.

Consider the following 4×4 image (colored marks denote places where a split can occur):

This image can be described by several sequences, for example : "<strong><span style="color: red">0</span></strong><strong><span style="color: blue">0</span></strong>10101010<strong><span style="color: green">0</span></strong>1011111011<strong><span style="color: orange">0</span></strong>10101010", of length 30, or "<strong><span style="color: red">0</span></strong>10<strong><span style="color: green">0</span></strong>101111101110", of length 16, which is the minimal sequence for this image.

For a positive integer $N$, define $D_N$ as the $2^N×2^N$ image with the following coloring scheme:

• the pixel with coordinates $x = 0$, $y = 0$ corresponds to the bottom left pixel,
• if ${(x - 2^{N - 1})}^2 + {(y - 2^{N - 1})}^2 ≤ 2^{2N - 2}$ then the pixel is black,
• otherwise the pixel is white.

What is the length of the minimal sequence describing $D_{24}$?

--hints--

quadtreeEncoding() should return 313135496.

assert.strictEqual(quadtreeEncoding(), 313135496);

--seed--

--seed-contents--

function quadtreeEncoding() {

  return true;
}

quadtreeEncoding();

--solutions--

// solution required