Let [z] mean the Greatest Integer less than or equal to z. Find a positive real number X, such that [X^n] is an even number whenever n is even, and [X^n] is an odd number whenever n is odd.
(In reply to
re(5): No Subject by Steve Herman)
I just haven't had much time of late to devote to perplexus puzzles - I'll just outline my general strategy and let others get the details:
1. Write a quadratic with integer coefficients: ax^2 + bx + c. Pick it so that the two roots r and s satisfy -1 < r < 0 < 1 < s.
2. Then, the sequence X[n] = r^n + s^n satisfies the recurrence aX[n+2] + bX[n+1] + cX[n] = 0.
3. X[n] will be an integer sequence. If a,b,c are chosen correctly, X[n] will always be even.
4. Since r^n oscillates sign with absolute value less than 1, [s^n] will oscillate parity.
Not just yet
