What sequence is defined by the string below, and more specifically, how is the string generated?
As a bonus, what is the next set of digits in the sequence?
(Hint: There are 21 of them.)
Whenever I think of rabbits in a Sequences context, I think of Fibonacci numbers. Whenever I see a string of 0's and 1's I think of binary numbers. We see here Fibonacci numbers in binary.
The sequence presented starts with F(1) = 1 and F(2) = 1, in binary. Each binary Fibonacci number F(n) is padded with enough leading zeroes to make the length of its representation F(n) digits long.
The following program produces the first 11 individual sets of digits:
a = 1: b = 1: PRINT "1": PRINT "1"
FOR fibNo = 3 TO 11
c = a + b
a = b: b = c
bn$ = ""
FOR i = 1 TO b
dig = c MOD 2
c = c \ 2
bn$ = LTRIM$(STR$(dig)) + bn$
IF c > 0 THEN END
So the last one presented in the puzzle is 0000000001101, and the next set, indeed with 21 digits, is 000000000000000010101.
You could even say that the sequence begins with F(0) = 0 presented with zero of its digits.
Posted by Charlie
on 2011-09-07 15:30:43