It is worth noting that N-bonacci sequences have a very practical application. They form the basis of what are called FIR (finite impulse response) digital filters. These are very widely used in applications to process signals. For example your mobile phone has many digital filters in it and these make the sound much better, so that you can talk to someone a long way away without interference. The performance of the digital filter is studied using the Z-transform which is a generalisation of the methods used to give equation (1) above. The infinacci sequence arises when you study IIR or inifinite impulse response digital filters. Interstingly these can be generated without using an inifinite number of terms, but that is another (quite long) story.
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