Complete Sequence

A sequence of numbers V={nu_n} is complete if every positive integer n is the sum of some subsequence of V, i.e., there exist a_i=0 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 the element {1} prepended,


is complete, even if any number of primes each >7 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 postulate.

See also

Bertrand's Postulate, Brown's Criterion, Fibonacci Dual Theorem, Greedy Algorithm, Weakly Complete Sequence, Zeckendorf's Theorem

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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.

Referenced on Wolfram|Alpha

Complete Sequence

Cite this as:

Weisstein, Eric W. "Complete Sequence." From MathWorld--A Wolfram Web Resource.

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