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- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Combinatorics_and_Graph_Theory_(Guichard)/03%3A_Generating_Functions/3.05%3A_Recurrence_RelationsA recurrence relation defines a sequence by expressing a typical term in terms of earlier terms. Note that some initial values must be specified for the recurrence relation to define a unique sequence...A recurrence relation defines a sequence by expressing a typical term in terms of earlier terms. Note that some initial values must be specified for the recurrence relation to define a unique sequence. The starting index for the sequence need not be zero if it doesn't make sense or some other starting index is more convenient.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji)/01%3A_Introduction/1.07%3A_Lame's_TheoremIn this section, we give an estimate to the number of steps needed to find the greatest common divisor of two integers using the Euclidean algorithm. To do this, we have to introduce the Fibonacci num...In this section, we give an estimate to the number of steps needed to find the greatest common divisor of two integers using the Euclidean algorithm. To do this, we have to introduce the Fibonacci numbers for the sake of proving a lemma that gives an estimate on the growth of Fibonacci numbers in the Fibonacci sequence. The lemma that we prove will be used in the proof of Lame’s theorem.
- https://math.libretexts.org/Courses/Mount_Royal_University/Mathematical_Reasoning/3%3A_Number_Patterns/3.3%3A__Recognising_SequencesAccording to the legend of the Tower of Hanoi (formerly the "Tower of Brahma" in a temple in the Indian city of Benares), the temple priests are to transfer a tower consisting of 64 fragile disks of g...According to the legend of the Tower of Hanoi (formerly the "Tower of Brahma" in a temple in the Indian city of Benares), the temple priests are to transfer a tower consisting of 64 fragile disks of gold from one part of the temple to another, one disk at a time. The number of moves needed to transfer n disks from post-A to post C is 2M+1, where M is the number of moves needed to transfer n−1 disks from post A to post C.
