Hey thanks Fred! Your mind-reader machine managed to figure out how the word "asymptotic" applies to Dan's riddle. All I got was a recurrence relation, which of course is not closed-form. But I was stumped by "asymptotic". But since these are Lucas numbers, then course, there must be an *exact*, closed form formula too (just as with the Fibonacci numbers) viz.: ((3-sqrt(5))^n + (3+sqrt(5))^n)/2^n This formula was contributed to A005248 by Creighton Dement back in Apr 19 2005. Remember "Floretions"? So we have a closed-form, exact not asymptotic, answer. On 1/31/13, Fred W. Helenius <fredh@ix.netcom.com> wrote:
[...]
2, 3, 7, 18, 47, 123, 322, 843, 2207, 5778, 15127, ...
This is the subsequence of the Lucas sequence consisting of the terms with even index. Since L(n) = phi^n + (-1/phi)^n, f(n) = L(2n) is asymptotic to phi^{2n}, where phi is the golden ratio.
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