Натуральные числа записали в бесконечную таблицу по спирали, начиная с единицы так, как...

# # Код на ruby 2.2.3p173
def LineUpArraySize(num)
   cur_size = 4
   while num > cur_size
       cur_size = 2 * cur_size + 8
   end
   cur_size = 2 * cur_size + 8
end

def getSpiralNeighbours(num)
   a = []
   a[0] = [0, 0]
   a[1] = [-1, 1]
   moves = [[1, 0], [0, -1], [-1, 0], [0, 1]]
   move_direction = 0
   side_size = 2
   cur_size = 1
   cycle = 1
   mode_direction = 0
   for i in 2..LineUpArraySize(num)
       new_num = Array.new(2)
       new_num[0] = a.last[0] + moves[move_direction][0]
       new_num[1] = a.last[1] + moves[move_direction][1]
       a[i] = new_num
       # p new_num
       cur_size += 1

       if cur_size == side_size
           cur_size = 1
           move_direction = (move_direction + 1) % 4
           side_size += 1 if move_direction % 2 == 0
           cycle += 1 if move_direction == 0
       end
   end

   a.each_with_index do |i,x|
       p [x, i] if (i[0]-a[num][0]).abs <= 1 and (i[1]-a[num][1]).abs <= 1<br>   end
end

getSpiralNeighbours(1370)

Вывод (числа и условные координаты)
[1226, [17, 18]]
[1369, [17, 19]]
[1370, [18, 19]]
[1371, [18, 18]]
[1520, [17, 20]]
[1521, [18, 20]]
[1522, [19, 20]]
[1523, [19, 19]]
[1524, [19, 18]]