93.45 Pseudo Fibonacci type sequences and their golden

摘要

Readers will be familiar with the manner in which the golden meanequation can be derived by examining the limiting case of ratios of terms inthe Fibonacci sequence. The purpose of this short note is to show how thisequation can be viewed as a special case of a more general one derived fromexamining, in an identical fashion, the limiting ratio of terms in a somewhatmore general sequence. Specifically, a sequence of a type not dissimilar tothe Fibonacci one is envisaged wherein the first n terms are just the integers0, 1, 2, 3, ... , n — 1 and thereafter each term of the sequence is simply thesum of the n terms that immediately precede it. Of course, in the case whenn = 2 the Fibonacci sequence is obtained. Thus, the terms of the proposedsequence, F (n) say, may be written down formally as follows: