*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)

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

**pa/pb (+) c/d = (pa+c)/(pb+d)**

a/b (-) qc/qd = (a-qc)/(b-qd)

a/b (-) qc/qd = (a-qc)/(b-qd)

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 |