THE LUCAS NUMBERS Fibonacci's companion — same rule, different seed, and L₅ = 11
Take the Fibonacci rule — each term is the sum of the two before — but start 2, 1 instead of 0, 1. You get the Lucas numbers: 2, 1, 3, 4, 7, 11, 18, 29, 47, 76, 123…. They are Fibonacci's twin — the other solution the recurrence admits — and they converge to the very same φ. They interlock with Fibonacci through dozens of identities (L(n) = F(n−1)+F(n+1), F(2n) = F(n)L(n)) and give the closed forms their integer backbone. The fifth Lucas number is 11 — the first piece of the puzzle.
The demo generates the Lucas sequence and marks L(5) = 11 — same recurrence as Fibonacci, seeded differently: live demo
HISTORY & CREDIT Édouard Lucas, 1870s
“Fibonacci is the sequence for that rule.” — the rule ‘sum the last two’ has a whole family of solutions; Lucas is the companion, and the two together are the natural basis. cited
1870s · Édouard Lucas — studied the sequence 2,1,3,4,7,11,… and named the sequence after Fibonacci (c.1876; the nickname “Fibonacci” for Leonardo of Pisa himself is Guillaume Libri's, 1838); his primality tests (Lucas–Lehmer) still guard the largest known primes. the bond · L(n)² − 5F(n)² = 4(−1)ⁿ; F(2n) = F(n)L(n). the 11 · L(5) = 11 — one of the referents this batch pieces together.
Same rule, second seed, same limit — and an 11 sitting fifth in line. The twin that makes Fibonacci a family. Lucas 1870s
RECOMMEND FOR I-13 the Lucas sequence, on the compiler
On the canonical compiler, L(5) = 11 and L(10) = 123 — the companion recurrence, seeded 2, 1:
$ i13 run g_lucas.i13 # L(n) = L(n-1)+L(n-2), seeds 2,1
RUN OK · 2981 step(s) · peak stack 4 · call depth 10
L5 = 11 -- the ~~11~~ referent
L10 = 123
Recommend: the Lucas numbers are Fibonacci's companion — the recurrence's other solution, converging to the same φ — and i13 grows them to L(5) = 11, the first of David's referents. Not a keeper (a second seed of the same rule is not a new mechanism), but the piece that shows “add the last two” is a family, not a single sequence, and the integer scaffold under Binet's closed form (dart 345).