Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's

A003714 on the OEIS · family binary rule · also known as Fibbinary numbers

Weight–level plate of Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's — Click the plate to explore it in 3-D. Weight k across, level L up, both on log scales: blue in the weight class (k ≤ L), orange in the level class (k > L). The dashed diagonal is k = L.

Open in the 3-D viewer A003714 on the OEIS

Terms	100,000 (n = 0 … 99,999)
Decomposable (a > 2d)	99,976
Level class, k > L	10,351 · 10.35 %
Weight class, k ≤ L	89,625 · 89.65 %
Ties, k = L	1
On the level line L = 1	5,372
Forced level, l ≤ d²	241
Range of a(n)	0 … 9,704,020
Range of the jump d	1 … 2,796,203
Largest weight k, level L	9,703,763, 4,852,008

No two adjacent 1 bits in binary. The decomposition fails 24 times, each where the sequence jumps to the next power of two (a(n+1) >= 1.5 a(n)). Gap 1 carries 62 % of the terms and the level share is 10.35 %.

The decomposition

Every term of a strictly increasing sequence is written a(n) = k(n)·L(n) + d(n): the jump d = a(n+1) − a(n), the weight k the least divisor of a − d greater than d, the level L = (a − d)/k. A term decomposes when a > 2d; it is in the level class when k > L and in the weight class when k ≤ L. See decompwlj.com and arXiv:0711.0865, or how it works, with worked examples and a live one.