A sequence of numbers is complete if every positive
is the sum of some subsequence of , i.e., there exist or 1 such that
(Honsberger 1985, pp. 123-126). The Fibonacci numbers are complete. In fact, dropping one number still leaves a complete sequence,
although dropping two numbers does not (Honsberger 1985, pp. 123 and 126). The
sequence of primes with
is complete, even if any number of primes each are dropped, as long as the dropped
terms do not include two consecutive primes (Honsberger
1985, pp. 127-128). This is a consequence of Bertrand's
Brown, J. L. Jr. "Unique Representations of Integers as Sums of Distinct Lucas Numbers." Fib. Quart.7,
243-252, 1969.Hoggatt, V. E. Jr.; Cox, N.; and Bicknell, M.
"A Primer for Fibonacci Numbers. XII." Fib. Quart.11, 317-331,
1973.Honsberger, R. Mathematical
Gems III. Washington, DC: Math. Assoc. Amer., 1985.