Anyone reading this is surely familiar with the Fibonacci numbers, that very famous sequence defined by the recurrence:
And though it’s not typical, for our purposes here it will be convenient to redefine the sequence so that:
Using this later definition, which is simply a shift by one place, it can be shown for n>0 that the nth Fibonacci number corresponds to the number of bit strings of length n that do not contain the sub-string “11”. Thus the number of bits strings of length n that do contain the sub-string “11” is:
We can also define a Fibonacci-like sequence with three terms as follows:
And in general this notion can be extended to a k-term, or rather k-step, sequence where:
See Wolfram Math World’s great explanation for more about this, although they obviously do not shift the sequences as we have done here. Notice however, that when shifted, the k-step Fibonacci sequences presented at Math World begin with the sub-sequence:
To generalize our bit string example from before, it can be observed that the nth term of the k-step Fibonacci sequence corresponds to the number of bit strings of length n that do not contain a sub-string of length k that is composed of all ones.
The Fibonacci numbers can be further generalized by introducing a scaling factor m, where we define:
where the initial elements of the sequence fibk,m are:
corresponds to the regular old Fibonacci numbers.
It turns out that the nth term of fibk,m corresponds to the number of strings of length n using an alphabet of size m that do not have a sub-string of k consecutive ones, assuming of course that the character “1” is part of the alphabet. Conversely, the number of strings that do contain such a sub-string is equal to:
Another interesting bit about all of this concerns the the limit of fibk,m(n+1)/fibk,m(n) as n goes to infinity. Here again fib2,2 has a special place as:
is a root of the polynomial:
Here is some Python code for playing with all of this.
#!/usr/bin/python import re import itertools def i2bs(i, l, m): """ Convert an incoming number into a 'bitstring' representation of length 'l' using an alphabet of size 'm'. This behavior of this function is undefined if m>10. """ i = int(i) out = "" while i: out = str(i%m) + out i /= m delta = l - len(out) if delta > 0: out = ("0" * delta) + out return out def fib(n, k=2, m=2): """ Generate the nth k-step, m-scale fibonacci number. n specifies which index in the sequence to generate. (length of the over all string) k specifies how many terms the recurrence has (length of the subsequence we are searching) m is the factor that each term of the reccurrence is multiplied by plus 1 (size of alphabet) The output is the nth number in the sequence, which is also equal to the number of times the subsequence "1"*k appears in the permutations of length n over the alphabet. """ f = [pow(m,i) for i in range(k)] if n < len(f): return f[n] i=k while i < n: f[i%k]=sum(f)*(m-1) i+=1 return sum(f)*(m-1) def matchbits(k,m): """ This function allows us to play with the proposition that fib(n,k,m) is the number of substrings "1"*k in the permutations of length n over an alphabet of size m. k is the size of substring to match m is the size of the alphabet """ print("***",k,m) ss=k*"1" for n in range(k,10): matches=0 total=pow(m,n) for i in range(total): x = i2bs(i,n,m) if re.search(ss,x): matches += 1 f =fib(n,k,m) print(("Subsequence=%s, Actual Matches=%d, Predicted Matches=%d, Okay=%s") % (ss, matches, total-f, ((total-f)== matches))) def poly(k,m,x): """ Evaluate the polynomial f(x) = x^k - (m-1)*x^(k-1) - ... - (m-1) """ return x**k + sum([-(m-1)*x**i for i in range(k)]) def checkroots(n, coefficients): """ Allows us to play with the polynomials of which fib(n+1, k, m)/fib(n, k, m) is a root. """ for (k,m) in itertools.product(coefficients, coefficients): fn = fib(n,k,m) fn1 = fib(n+1,k,m) x=float(fn1)/float(fn) print("k=%d, m=%d, x=%f, f(x)=%f") % (k, m, x, poly(k,m, x)) def main(): print("Check Roots") checkroots(100, [2,3,4,5]) print("\n") print("Match Bits:") matchbits(2,2) #matchbits(3,2) #matchbits(3,4) #matchbits(5,4) if __name__ == "__main__": main();