This discussion and my own (preliminary) benchmarking results have led me to rethink a few things about my real problem.

For instance, when computing F(n), we can make use of the addition formula to break n into 2 pieces: 

F(n) = F(i+j) = F(i-1)*F(j) + F(i)*F(j+1)

[download]
Then: 
j = k**2 = int(sqrt(n))
i = n - j
F(n) = F(i+j) = F(i+k**2)

[download]
which we can compute using the first formula above and 
F(k**2) = (F(k-1)+F(k+1))*F(k*(k-1)) 
          - ((-1)**k)*F(k*(k-2))

[download]
It remains to be seen whether all of this extra work buys anything.

-QM
--
Quantum Mechanics: The dreams stuff is made of

In reply to Re^6: OT: Finding Factor Closest To Square Root by QM
in thread OT: Finding Factor Closest To Square Root by QM

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post, it's "PerlMonks-approved HTML":


  • Posts are HTML formatted. Put <p> </p> tags around your paragraphs. Put <code> </code> tags around your code and data!
  • Titles consisting of a single word are discouraged, and in most cases are disallowed outright.
  • Read Where should I post X? if you're not absolutely sure you're posting in the right place.
  • Please read these before you post! —
  • Posts may use any of the Perl Monks Approved HTML tags:
    a, abbr, b, big, blockquote, br, caption, center, col, colgroup, dd, del, details, div, dl, dt, em, font, h1, h2, h3, h4, h5, h6, hr, i, ins, li, ol, p, pre, readmore, small, span, spoiler, strike, strong, sub, summary, sup, table, tbody, td, tfoot, th, thead, tr, tt, u, ul, wbr
  • You may need to use entities for some characters, as follows. (Exception: Within code tags, you can put the characters literally.)
            For:     Use:
    & &amp;
    < &lt;
    > &gt;
    [ &#91;
    ] &#93;
  • Link using PerlMonks shortcuts! What shortcuts can I use for linking?
  • See Writeup Formatting Tips and other pages linked from there for more info.