Question

Let me explain with an example. If n=4 and r=2 that means all 4 digit binary numbers such that two adjacent digits can be 1. so the answer is 0011 0110 1011 1100 1101

Answer 1

Q. i am unable to figure out a pattern or an algorithm.

Hint: The 11 can start in position 0, 1, or 2. On either side, the digit must be zero, so the only “free” digits are in the remaining position and can cycle through all possible values.

For example, if there are n=10 digits and you’re looking for r=3 adjacent ones, the pattern is

x01110y   

Where x and y can cycle through all possible suffixes and prefixes for the remaining five free digits. Note, on the sides, the leading and trailing zero gets dropped, leaving six free digits in x0111 and 1110y.

Here’s an example using Python:

from itertools import product

def gen(n, r):
    'Generate all n-length sequences with r fixed adjacent ones'
    result = set()

    fixed = tuple([1] * r + [0])
    for suffix in product([0,1], repeat=n-r-1):
        result.add(fixed + suffix)

    fixed = tuple([0] + [1] * r + [0])
    rem = n - r - 2
    for leadsize in range(1, rem):
        for digits in product([0,1], repeat=rem):
            result.add(digits[:leadsize] + fixed + digits[leadsize:])

    fixed = tuple([0] + [1] * r)
    for prefix in product([0,1], repeat=n-r-1):
        result.add(prefix + fixed)

    return sorted(result)