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sage: for i in (1..5):
....: c = b+1
....: b = 1/c
....: print b.simplify_full(),",",c.simplify_full()
....:
a/(a + 1) , (a + 1)/a
(a + 1)/(2*a + 1) , (2*a + 1)/(a + 1)
(2*a + 1)/(3*a + 2) , (3*a + 2)/(2*a + 1)
(3*a + 2)/(5*a + 3) , (5*a + 3)/(3*a + 2)
(5*a + 3)/(8*a + 5) , (8*a + 5)/(5*a + 3)
That said, I'd use something like this:
sage: L = []
sage: for i in (1..5):
c = b+1
b = 1/c
L.append([b.simplify_full(),c.simplify_full()])
....:
sage: table(L,header_row=['b','c'])
b c
+-------------------------+-------------------------+
(8*a + 5)/(13*a + 8) (13*a + 8)/(8*a + 5)
(13*a + 8)/(21*a + 13) (21*a + 13)/(13*a + 8)
(21*a + 13)/(34*a + 21) (34*a + 21)/(21*a + 13)
(34*a + 21)/(55*a + 34) (55*a + 34)/(34*a + 21)
(55*a + 34)/(89*a + 55) (89*a + 55)/(55*a + 34)
In the notebook if you do html(table(...)) and make sure to have everything be in latex() like so
sage: for i in (1..5):
c = b+1
b = 1/c
L.append(['$'+latex(b.simplify_full())+'$','$'+latex(c.simplify_full())+'$'])
....:
sage: html(table(L,header_row=['b','c']))
<html>
<div class="notruncate">
<table class="table_form">
<tbody>
<tr>
<th>b</th>
<th>c</th>
</tr>
<tr class ="row-a">
<td><script type="math/tex"> \frac{144 \, a + 89}{233 \, a + 144} </script></td>
<td><script type="math/tex"> \frac{233 \, a + 144}{144 \, a + 89} </script></td>
</tr>
<tr class ="row-b">
<td><script type="math/tex"> \frac{233 \, a + 144}{377 \, a + 233} </script></td>
<td><script type="math/tex"> \frac{377 \, a + 233}{233 \, a + 144} </script></td>
</tr>
<tr class ="row-a">
<td><script type="math/tex"> \frac{377 \, a + 233}{610 \, a + 377} </script></td>
<td><script type="math/tex"> \frac{610 \, a + 377}{377 \, a + 233} </script></td>
</tr>
<tr class ="row-b">
<td><script type="math/tex"> \frac{610 \, a + 377}{987 \, a + 610} </script></td>
<td><script type="math/tex"> \frac{987 \, a + 610}{610 \, a + 377} </script></td>
</tr>
<tr class ="row-a">
<td><script type="math/tex"> \frac{987 \, a + 610}{1597 \, a + 987} </script></td>
<td><script type="math/tex"> \frac{1597 \, a + 987}{987 \, a + 610} </script></td>
</tr>
</tbody>
</table>
</div>
</html>
could be even more awesome. Good luck!
PS naturally I didn't answer about polynomial fields, but this is also possible, I think. See e.g. here.
