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Explicit Formula for the Fibonacci & Lucas Numbers
Only a strong foundation in precalculus, plus a good background in matrices, determinants, congruences, and combinatorics is required. The text may be used in a variety of number theory courses, as well as in seminars, workshops, and other capstone experiences for teachers in-training and instructors at all levels.
Pell and Pell–Lucas Numbers with Applications
Lucas numbers and Fibonacci numbers form complementary instances of Lucas sequences. The Lucas sequence has the same recursive relationship as the Fibonacci sequence , where each term is the sum of the two previous terms, but with different starting values. This produces a sequence where the ratios of successive terms approach the golden ratio , and in fact the terms themselves are roundings of integer powers of the golden ratio. Similar to the Fibonacci numbers, each Lucas number is defined to be the sum of its two immediate previous terms, thereby forming a Fibonacci integer sequence. Though closely related in definition, Lucas and Fibonacci numbers exhibit distinct properties. All Fibonacci-like integer sequences appear in shifted form as a row of the Wythoff array ; the Fibonacci sequence itself is the first row and the Lucas sequence is the second row. Also like all Fibonacci-like integer sequences, the ratio between two consecutive Lucas numbers converges to the golden ratio.
The sequence now known as Fibonacci numbers sequence 0, 1, 1, 2, 3, 5, 8, Pingala's work with the mountain of cadence now known as Pascal's triangle made him the first known person to have looked into Fibonacci numbers. Next, another Indian mathematician, Virahanka 6th century AD , took note of the Fibonacci sequence through analysis of a completely different problem. Virahanka considered the following problem: assuming that lines of n units are composed of syllables that can be long or short—a long syllable takes twice as long as a short syllable to articulate—and each line of n units takes the same time to articulate no matter how it is composed, how many different combinations of syllables are there for each line of length n? Research of this question was continued by the Indian scholar Hemachandra and the Indian mathematician Gopala in the 12th century.
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