Lucas number

Not to be confused with Lucas sequences, the general class of sequences to which the Lucas numbers belong.
The Lucas spiral, made with quarter-arcs, is a good approximation of the golden spiral when its terms are large. However, when its terms become very small, the arc's radius decreases rapidly from 3 to 1 then increases from 1 to 2.

The Lucas sequence is an integer sequence named after the mathematician François Édouard Anatole Lucas (1842–1891), who studied both that sequence and the closely related Fibonacci sequence. Individual numbers in the Lucas sequence are known as Lucas numbers. 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.[1] 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.[2] The sequence also has a variety of relationships with the Fibonacci numbers, like the fact that adding any two Fibonacci numbers two terms apart in the Fibonacci sequence results in the Lucas number in between.[3]

The first few Lucas numbers are

2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364, 2207, 3571, 5778, 9349, ... . (sequence A000032 in the OEIS)

which coincides for example with the number of independent vertex sets for cyclic graphs of length .[1]

  1. ^ a b Weisstein, Eric W. "Lucas Number". mathworld.wolfram.com. Retrieved 2020-08-11.
  2. ^ Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 284. ISBN 978-0-374-53563-6.
  3. ^ Parker, Matt (2014). "13". Things to Make and Do in the Fourth Dimension. Farrar, Straus and Giroux. p. 282. ISBN 978-0-374-53563-6.