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Next to the factor, two successive terms of a series of the type U(a,b)F are needed to develop it. To develop all series involved in an approach, one needs two successive sections. Suppose a program peters out before completing the second section. In the worst case scenario it just managed the first non-trivial sp-block:
 1 0 1 2 3 4 7 11 18 83 101 119 137 256 393 649 2340 ? ? ? ? ? ? ? ? ? ? ? ? ? 842401 3037320 ... 0 1 1 1 1 1 2 3 5 23 28 33 38 71 109 180 649 ? ? ? ? ? ? ? ? ? ? ? ? ? 233640 842401 ... 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 ...

Top- and middle row give the numerators and denominators of the successively better rational approaches of root 13. The bottom row gives the index. As far as root approach goes, we can simply develop the sp-series. But if, for some theoretical reason, we would want all series, it's good to know that all sections show the same 'incremental profile', so that it is possible to obtain the second section (and therewith all series) from the first one's profile.

Let's define the operations to 'increment' or 'decrement' one fraction with another as follows:

a/b (+) c/d = (a+c)/(b+d)
a/b (-) c/d = (a-c)/(b-d)

An identical fraction in a different representation, renders a different result:

pa/pb (+) c/d = (pa+c)/(pb+d)
a/b (-) qc/qd = (a-qc)/(b-qd)

p and q act as parameters of the 'weight' in the outcome, of a/b an c/d respectively.

A next term in a section can always be obtained by in- or decrementing the last term with previous ones, including the last one itself. All sections show the same pattern, called the 'incremental profile'.
To simplify things, we will indicate fractions by their index and increment the second section according to the first one's profile. The increment within the ab-block is disregarded, so there are 14 increments, starting on the Nb/a-fraction, of which the last one should render the next non-trivial a/b-fraction 842401/233640. Since we know that one already, it's a way to check the result.
• | i | means: increment the last fraction with the fraction on index i.
• | -i | means: decrement likewise.
• | p*i,-j | means: increment p times with the fraction indexed i and decrement with the fraction indexed j.

The profile of the first section looks like this:
0/1: | 0 | 0 | 0 | 0 | 4 | 5 | 6 | 3*8,7 | 8 | 8 | 8 | 11 | 12 | 13 |

or alternatively:
0/1: | 0 | 0 | 0 | 0 | 4 | 5 | 6 | 4*8,-6 | 8 | 8 | 8 | 11 | 12 | 13 |

To get the second section, take the indices modulo 15
2340/649: | 15 | 15 | 15 | 15 | 19 | 20 | 21 | 3*23,22 | 23 | 23 | 23 | 26 | 27 | 28 |

or alternatively:
2340/649: | 15 | 15 | 15 | 15 | 19 | 20 | 21 | 4*23,-21 | 23 | 23 | 23 | 26 | 27 | 28 |

The result should look like this:
 2340 2989 3638 4287 4936 9223 14159 23382 107687 131069 154451 177833 332284 510117 842401 649 829 1009 1189 1369 2558 3927 6485 29867 36352 42837 49322 92159 141481 233640 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30