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Объяснение:

9/2=4 (1); 4/2=2 (0); 2/2=1 (0); (1); 9₁₀=1001₂

34/2=17 (0); 17/2=8 (1); 8/2=4 (0); 4/2=2 (0); 2/2=1 (0); (1); 34₁₀=100010₂

59/2=29 (1); 29/2=14 (1); 14/2=7 (0); 7/2=3 (1); 3/2=1 (1); (1); 59₁₀=111011₂

629/2=314 (1); 314/2=157 (0); 157/2=78 (1); 78/2=39 (0); 39/2=19 (1); 19/2=9 (1);

9/2=4 (1); 4/2=2 (0); 2/2=1 (0); (1); 629₁₀=1001110101₂

936/2=468 (0); 468/2=234 (0); 234/2=117 (0); 117/2=58 (1); 58/2=29 (0);

29/2=14 (1); 14/2=7 (0); 7/2=3 (1); 3/2=1 (1); (1); 936₁₀=1110101000₂

1875/2=937 (1); 937/2=468 (1); 468/2=234 (0); 234/2=117 (0); 117/2=58 (1);

58/2=29 (0); 29/2=14 (1); 14/2=7 (0); 7/2=3 (1); 3/2=1 (1); (1); 1875₁₀=11101010011₂

3913/2=1956 (1); 1956/2=978 (0); 978/2=489 (0); 489/2=244 (1); 244/2=122 (0);

122/2=61 (0); 61/2=30 (1); 30/2=15 (0); 15/2=7 (1); 7/2=3 (1); 3/2=1 (1); (1);

3913₁₀=111101001001₂

11649/2=5824 (1); 5824/2=2912 (0); 2912/2=1456 (0); 1456/2=728 (0);

728/2=364 (0); 364/2=182 (0); 182/2=91 (0); 91/2=45 (1); 45/2=22 (1);

22/2=11 (0); 11/2=5 (1); 5/2=2 (1); 2/2=1 (0); (1); 11649₁₀=10110110000001₂

39578/2=19789 (0); 19789/2=9894 (1); 9894/2=4947 (0); 4947/2=2473 (1);

2473/2=1236 (1); 1236/2=618 (0); 618/2=309 (0); 309/2=154 (1); 154/2=77 (0);

77/2=38 (1); 38/2=19 (0); 19/2=9 (1); 9/2=4 (1); 4/2=2 (0); 2/2=1 (0); (1);

39578₁₀=1001101010011010₂

53746/2=26873 (0); 26873/2=13436 (1); 13436/2=6718 (0); 6718/2=3359 (0);

3359/2=1679 (1); 1679/2=839 (1); 839/2=419 (1); 419/2=209 (1); 209/2=104 (1);

104/2=52 (0); 52/2=26 (0); 26/2=13 (0); 13/2=6 (1); 6/2=3 (0); 3/2=1 (1); (1);

53746₁₀=1101000111110010₂

# # 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
    end
end

getSpiralNeighbours(1090)

Вывод (числа и условные координаты)
[962, [15, 16]]
[1089, [15, 17]]
[1090, [16, 17]]
[1091, [16, 16]]
[1224, [15, 18]]
[1225, [16, 18]]
[1226, [17, 18]]
[1227, [17, 17]]
[1228, [17, 16]]