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I asked this question on Mathoverflow before I realized that that was meant for research.

I'd like to find a closed form for the nth term of this sequence, much like there is the closed form for the Fibonacci numbers. There are two sequences in question, $f(n)$ and $g(n)$. They look like this:

$f(n): 1, 3, 7, 17, 41, 99, ...$

$g(n): 1, 2, 5, 12, 29, 70, ...$

They behave the same way, in that their rule is that for each series $s(n)$, $s(n)=2s(n-1)+s(n-2)$. The only difference between the two series is that $f(n)$ begins with 1 and 3, and $g(n)$ begins with 1 and 2.

There is one bit of knowledge I have. Don't know if it helps. But for each n, $g(n) / f(n)$ approaches $1/\sqrt2$. Not sure whether this constant has any significance.

Inspired by Mathologer. Thank you for the help.

J. W. Tanner
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Kaspar Poland
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1 Answers1

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The recurrence $s_n=2s_{n-1}+s_{n-2}$ can be solved

by solving the characteristic equation $r^2=2r+1$.

The solutions to that equation are $r=1\pm\sqrt2$.

The solutions to the recurrence are then $s_n=A(1+\sqrt2)^n+B(1-\sqrt2)^n$.

Can you solve for $A$ and $B$ for your two different initial conditions?

Can you see why $g(n)/f(n)$ approaches $1/\sqrt2$?

J. W. Tanner
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