Single moves

The board is an infinite 2 dimensional square grid, like a limitless chess board. A piece with value N (an N-mover) can move to any square that is a distance of exactly the square root of N from its current square (Euclidean distance measured centre to centre).

For example:

• A 1-mover can move to any square that is horizontally or vertically adjacent


• A 2-mover can move to any square that is diagonally adjacent


• A 5-mover moves like a chess knight

Note that not all N-movers can move. A 3-mover can never leave its current square because none of the squares on the board are a distance of exactly root 3 from the current square.

Multiple moves

If allowed to move repeatedly, some pieces can reach any square on the board. For example, a 1-mover and a 5-mover can both do this. A 2-mover can only move diagonally and can only reach half of the squares. A piece that cannot move, like a 3-mover, cannot reach any of the squares (the starting square is not counted as "reached" if no movement occurs).

The images show which squares can be reached. More details on hover. Click for larger image.

• Squares reachable in 1 or more moves are marked in black


• Squares reachable in exactly 1 move are shown by red pieces
  (apart from the 3-mover, which cannot move)
  

What proportion of the board can a given N-mover reach?

Input

• A positive integer N

Output

• The proportion of the board that an N-mover can reach


• This is a number from 0 to 1 (both inclusive)


• For this challenge, output as a fraction in lowest terms, like 1/4, is allowed

So for input 8, both 1/8 and 0.125 are acceptable outputs.

Scoring

This is code golf. The score is the length of the code in bytes. For each language, the shortest code wins.

Test cases

In the format input : output as fraction : output as decimal

  1 : 1     : 1

  2 : 1/2   : 0.5

  3 : 0     : 0

  4 : 1/4   : 0.25

  5 : 1     : 1

  6 : 0     : 0

  7 : 0     : 0

  8 : 1/8   : 0.125

  9 : 1/9   : 0.1111111111111111111111111111

 10 : 1/2   : 0.5

 13 : 1     : 1

 16 : 1/16  : 0.0625

 18 : 1/18  : 0.05555555555555555555555555556

 20 : 1/4   : 0.25

 25 : 1     : 1

 26 : 1/2   : 0.5

 64 : 1/64  : 0.015625

 65 : 1     : 1

 72 : 1/72  : 0.01388888888888888888888888889

 73 : 1     : 1

 74 : 1/2   : 0.5

 80 : 1/16  : 0.0625

 81 : 1/81  : 0.01234567901234567901234567901

 82 : 1/2   : 0.5

144 : 1/144 : 0.006944444444444444444444444444

145 : 1     : 1

146 : 1/2   : 0.5

148 : 1/4   : 0.25

153 : 1/9   : 0.1111111111111111111111111111

160 : 1/32  : 0.03125

161 : 0     : 0

162 : 1/162 : 0.006172839506172839506172839506

163 : 0     : 0

164 : 1/4   : 0.25

241 : 1     : 1

242 : 1/242 : 0.004132231404958677685950413223

244 : 1/4   : 0.25

245 : 1/49  : 0.02040816326530612244897959184

260 : 1/4   : 0.25

261 : 1/9   : 0.1111111111111111111111111111

288 : 1/288 : 0.003472222222222222222222222222

290 : 1/2   : 0.5

292 : 1/4   : 0.25

293 : 1     : 1

324 : 1/324 : 0.003086419753086419753086419753

325 : 1     : 1

326 : 0     : 0

360 : 1/72  : 0.01388888888888888888888888889

361 : 1/361 : 0.002770083102493074792243767313

362 : 1/2   : 0.5

369 : 1/9   : 0.1111111111111111111111111111

370 : 1/2   : 0.5

449 : 1     : 1

450 : 1/18  : 0.05555555555555555555555555556

488 : 1/8   : 0.125

489 : 0     : 0

490 : 1/98  : 0.01020408163265306122448979592

520 : 1/8   : 0.125

521 : 1     : 1

522 : 1/18  : 0.05555555555555555555555555556

544 : 1/32  : 0.03125

548 : 1/4   : 0.25

549 : 1/9   : 0.1111111111111111111111111111

584 : 1/8   : 0.125

585 : 1/9   : 0.1111111111111111111111111111

586 : 1/2   : 0.5

592 : 1/16  : 0.0625

593 : 1     : 1

596 : 1/4   : 0.25

605 : 1/121 : 0.008264462809917355371900826446

610 : 1/2   : 0.5

611 : 0     : 0

612 : 1/36  : 0.02777777777777777777777777778

613 : 1     : 1

624 : 0     : 0

625 : 1     : 1

Clean, 189 185 172 bytes

import StdEnv

$n#p=[[x,y]\x<-[~n..n],y<-[~n..n]|x^2+y^2==n]

=sum[1.0\_<-iter n(q=removeDup[k\[a,b]<-[[0,0]:p],[u,v]<-q,k<-[[a+u,b+v]]|all(e=n>=e&&e>0)k])p]/toReal(n^2)

Try it online!

Finds every position reachable in the n-side-length square cornered on the origin in the first quadrant, then divides by n^2 to get the portion of all cells reachable.