Just how far we could/should drive the intuition of the series operator is an interesting question.

I personally believe that however appealing at first sight, it would be kinda too much! When the layman, or even the Physics doctor asks: "which is the next term after 1,1,2,3,5,8,13,21?" expecting his audience either to recocgnize the Fibonacci numbers or not to know about them, and then to blow their minds with explanations about them... the mathematician would answer: "42" In fact, given any number one can devise a rule by which the latter can be obtained from the former ones. Of course, one would want an "easy" rule. But the problem is that that "easy" is just as much as psychologically appealing as difficult to be mathematically defined.

Thus, the question is either devoid of any sensible answer, or one should restrict himself a priori to some predefined class of rules, e.g. to linear recurrencies, and to establish an unambiguous relationship of simplicity within that class, to allow one to be picked up automatically.

Alternatively, or better, in addition, one may allow other classes to be specified by means of a suitable adverb, e.g.