Inmathematics, thefibbinary numbers are the numbers whosebinary representation does not contain two consecutive ones. That is, they are sums of distinct and non-consecutivepowers of two.^[1][2]

The fibbinary numbers were given their name by Marc LeBrun, because they combine certain properties ofbinary numbers andFibonacci numbers:^[1]

• The number of fibbinary numbers less than any given power of two is a Fibonacci number. For instance, there are 13 fibbinary numbers less than 32, the numbers 0, 1, 2, 4, 5, 8, 9, 10, 16, 17, 18, 20, and 21.^[1]
• The condition of having no two consecutive ones, used in binary to define the fibbinary numbers, is the same condition used in theZeckendorf representation of any number as a sum of non-consecutive Fibonacci numbers.^[1]
• The${\displaystyle n}$th fibbinary number (counting 0 as the 0th number) can be calculated by expressing${\displaystyle n}$ in its Zeckendorf representation, and re-interpreting the resulting binary sequence as the binary representation of a number.^[1] For instance, the Zeckendorf representation of 19 is 101001 (where the 1's mark the positions of the Fibonacci numbers used in the expansion19 = 13 + 5 + 1), the binary sequence 101001, interpreted as a binary number, represents41 = 32 + 8 + 1, and the 19th fibbinary number is 41.
• The${\displaystyle n}$th fibbinary number (again, counting 0 as 0th) iseven orodd if and only if the${\displaystyle n}$th value in theFibonacci word is 0 or 1, respectively.^[3]

Because the property of having no two consecutive ones defines aregular language, the binary representations of fibbinary numbers can be recognized by afinite automaton, which means that the fibbinary numbers form a2-automatic set.^[4]

The fibbinary numbers include theMoser–de Bruijn sequence, sums of distinct powers of four. Just as the fibbinary numbers can be formed by reinterpreting Zeckendorff representations as binary, the Moser–de Bruijn sequence can be formed by reinterpreting binary representations as quaternary.^[5]

A number${\displaystyle n}$ is a fibbinary number if and only if thebinomial coefficient${\displaystyle (\genfrac{}{}{0ex}{}{3n}{n})}$ is odd.^[1] Relatedly,${\displaystyle n}$ is fibbinary if and only if the centralStirling number of the second kind${\displaystyle \left\{\genfrac{}{}{0ex}{}{2n}{n}\right\}}$ is odd.^[6]

Every fibbinary number${\displaystyle f_i}$ takes one of the two forms${\displaystyle 2f_j}$ or${\displaystyle 4f_j+1}$, where${\displaystyle f_j}$ is another fibbinary number.^[3][7]Correspondingly, thepower series whose exponents are fibbinary numbers,$${\displaystyle B(x)=1+x+x^2+x^4+x^5+x^8+\cdots ,}$$obeys thefunctional equation^[2]$${\displaystyle B(x)=xB(x^4)+B(x^2).}$$

Madritsch & Wagner (2010) provide asymptotic formulas for the number ofinteger partitions in which all parts are fibbinary.^[7]

If ahypercube graph${\displaystyle Q_d}$ of dimension${\displaystyle d}$ is indexed byintegers from 0 to${\displaystyle 2^d-1}$, so that twovertices are adjacent when their indexes have binary representations withHamming distance one, then thesubset of vertices indexed by the fibbinary numbers forms aFibonacci cube as itsinduced subgraph.^[8]

Every number has a fibbinary multiple. For instance, 15 is not fibbinary, but multiplying it by 11 produces 165 (10100101₂), which is.^[9]

• Binary arithmetic
• Base-dependent integer sequences
• Fibonacci numbers

• Articles with short description
• Short description is different from Wikidata