Continuant, caterpillar, and topological index Z. Fastest algorithm for degrading a continued fraction

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紀要論文

Manipulation of continued fraction, either finite and infinite, was shown to be greatly simplified and systematized by introducing the topological index Z and caterpillar graph. The continuant which was introduced by Euler in 18 century for solving continued fraction problems was shown to be identical to the Z-index of the caterpillar graph derived from the continued fraction concerned. Then the fastest algorithm for solving the Pell equations was obtained. Further, graph-theoretical interpretation for Fibonacci and Lucas numbers, and generalized Fibonacci numbers was obtained.

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