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Recently, I've been playing a lot with Brown's Criterion and writing numbers as a sum of members of a sequence. However, I have observed that Brown's Criterion does not seem to be entirely accurate for some sequences.

For example, take the Tribonacci sequence ($x_n = x_{n-1} + x_{n-w} + x_{n-3}$) and apply it for initial conditions $x_1 = 1, x_2 = 1, x_3 = 1$. The first few terms are listed below:

$$ 1, 1, 1, 3, 5, 9, 17, 31, 57 $$

Because this is a Tribonacci sequence, the ratio between terms as the number of members approaches infinity is approximately 1.839, which is less than 2. For all terms except one, Brown's Criterion is met. The term which does not follow Brown's Criterion is the fourth term, 3, as it is is greater than two times the previous term. Yet I can still express every positive integer as a sum of members of this sequence.

Am I wrong? And, if not, are there other exceptions to Brown's Criterion?

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    The criterion is "if" i.e. it guarantees completeness, but not "only if" - it is not necessary for completeness. If I begin with a long sequence of $1$s I can continue $3,9,27 \dots$, for example, and complete by adding at each stage the first integer not yet represented. Say $39=3+9+27$ ones to be sure. I can also do something like $1,1,1,3,7,13,27 \dots$ where I introduce a little redundancy to get alternating rations greater than $2$ – 2017-02-07
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    @MarkBennet The material I had seen stated that it was an "iff" relationship for completeness. :\ Guess that's a source problem. EDIT: Actually, _all_ of the sources I have seen state that it's an "iff" relationship. Is there a better source for this subject? – 2017-02-07
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    From what I see in the references, the condition $x_{n+1}\le2\,x_n$ is a sufficient but not necessary condition for completeness of he sequence $\{x_n\}$. It is a corollary of Brown´s criterion, $\sum_{k=1}^{n-1}x_k\le x_n-1$. – 2017-02-08

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