486 Mb WARBURTON, ON SELF-REPEATING SERIES. 



terms of the triangular series 1, 3, 6, 10, &c, whose generating fraction is - — , we have the 



(I — t) 



self-repeating series of the square numbers, the zero-terms being reduced from two to a single 



term. 



The generating fraction is then, — . 



If we add together every three consecutive terms, we have the series, 1, 4, 10, 19, 31, &c, 

 from which all the zero-terms are obliterated ; the generating fraction being in this case 



1 + t + f 



(i - ty ' 



14. But what, it may be asked, would be the result, if having by previous addition obli- 

 terated all the zero-terms included between the right and the left arm, we were to add together 

 every (n + l), or (ra + 2), or more, consecutive terms of the series. My answer is, that we 

 should obtain an improper fraction whose numerator and denominator would be recurrent ; and 

 we might then possibly have to consider the case of a self-repeating series, differing from those 

 of which I have been treating, in its containing between the two arms a finite middle term. 



A theorem respecting this second class of self-repeating series, analogous to that which 

 applies to the first class of which we have been treating, I shall make the subject of a future 

 communication. 



HENRY WARBURTON. 



May 15, 1854. 



