Now you can vary the parameter a very quickly. So far

is the only one that converges to an integer.

]]>Have you tried the next one?

]]>Thanks for the reply. The surprise for me is all those odd powers of an irrational conspiring to yield an integer! Putting the next phi-like term in the series, that is (3 + √5)/2, for n leaves the convergence unchanged at 2 x phi but now yields 3 when truncated to only 2 terms, leaving a somewhat more trivial algebra of (3 + √5)/2 + 2/(3 + √5) = 3.

]]>That is a very good piece of experimental math.

a) I did not know that before.

b) Sorry, my field is numerical analysis, so I am what they call a thrasher. Geometric ideas rarely penetrate my skull But there is an algebraic reason for your discovery.

On its way to becoming:

Your sum briefly stops at exactly 3. That is the only integer it ever touches.

If we substitute phi into your finite sum of

We get:

]]>a) Anyone's ever noted this before

b) There's anything deeper here, such as a geometric explanation