From Osher Doctorow
Lucas Numbers Ln (see Wolfram's "Lucas Numbers" for a simple but
useful introduction) are just like Fibonacci Numbers Fn in their
defining additive equation except that L1 = 1, L2 = 3:
1) Ln = Ln-1 + Ln-2, L1 = 1, L2 = 3
They have several somewhat similar relationships to the Golden Ratio
(somewhat similar to relationships of Fibonacci Numbers), but there
are usually some changes. There are many fascinating equations
relating Lucas and Fibonacci Numbers to each other and to other
quantities.
Aren't Fibonacci and Lucas Numbers rather rare, and doesn't this
counteract the idea that addition (as of positive integers) is built
up from them? Actually, they have surprising "fill in" behavior with
regard to positive integers and primes, generating a large number of
the primes although not all. Fibonacci numbers are 1, 1, 2, 3, 5, 8,
13, 21, ..., which seems to leave primes 7 and 11 out, but Lucas
numbers are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, ..., which "fill in"
the 7 and 11 and the non-prime 4 as well, etc.
Osher Doctorow
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