Complete sequence

In mathematics, an integer sequence is called a complete sequence if every positive integer can be expressed as a sum of values in the sequence, using each value at most once.

For example, the sequence of powers of two {1, 2, 4, 8, ...}, based on the binary numeral system, is a complete sequence; given any natural number, we can choose the values corresponding to the 1 bits in its binary representation and sum them to obtain that number (e.g. 37 = 100101₂ = 1 + 4 + 32). This sequence is minimal, since no value can be removed from it without making some natural numbers impossible to represent. Simple examples of sequences that are not complete include:

• The even numbers; since adding even numbers produces only even numbers, no odd number can be formed.
• Powers of three; no integer having a digit "2" in its ternary representation (2, 5, 6...) can be formed.

Without loss of generality, assume the sequence a_n is in nondecreasing order, and define the partial sums of a_n as:

$s_n=\sum_{m=0}^n a_m$.

Then the conditions

a₀ = 1

$s_{k-1} \ge a_k - 1 \, \forall \, k \ge 1$

are both necessary and sufficient for a_n to be a complete sequence.^[1]

Other complete sequences include:

• The sequence of the number 1 followed by the prime numbers; this follows from Bertrand's postulate.^[1]
• The Fibonacci numbers, as well as the Fibonacci numbers with any one number removed.^[1] This follows from the identity that the sum of the first n Fibonacci numbers is the (n + 2)nd Fibonacci number minus 1 (see Fibonacci_numbers#Second_identity).

Applications

Just as the powers of two form a complete sequence due to the binary numeral system, in fact any complete sequence can be used to encode integers as bit strings. The rightmost bit position is assigned to the first, smallest member of the sequence; the next rightmost to the next member; and so on. Bits set to 1 are included in the sum. These representations may not be unique.

For example, in the Fibonacci arithmetic system, based on the Fibonacci sequence, the number 17 can be encoded in six different ways:

110111 (F₆ + F₅ + F₃ + F₂ + F₁ = 8 + 5 + 2 + 1 + 1 = 17, maximal form)

111001 (F₆ + F₅ + F₄ + F₁ = 8 + 5 + 3 + 1 = 17)

111010 (F₆ + F₅ + F₄ + F₂ = 8 + 5 + 3 + 1 = 17)

1000111 (F₇ + F₃ + F₂ + F₁ = 13 + 2 + 1 + 1 = 17)

1001001 (F₇ + F₄ + F₁ = 13 + 3 + 1 = 17)

1001010 (F₇ + F₄ + F₂ = 13 + 3 + 1 = 17, minimal form, as used in Fibonacci coding)

In this numeral system, any substring "100" can be replaced by "011" and vice versa due to the definition of the Fibonacci numbers.^[2]

References

• Weisstein, Eric W., "Complete Sequence" from MathWorld.

1. ^ ^{a b c} Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., 1985, pp.123-128.
2. ^ Alexey Stakhov. "The main operations of the Fibonacci arithmetic". Museum of Harmony and Golden Section. http://www.goldenmuseum.com/1202FibCdeTransf_engl.html. Retrieved 27 July 2010.