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Statement

Setting (p. 1). For a finite or infinite sequence of integers $A=(a_1,a_2,\ldots)$, $P(A)$ is the set of integers $\sum_k\epsilon_ka_k$ with every $\epsilon_k\in\{0,1\}$, that is, the sums of finitely many distinct terms of $A$. The sequence $A$ is complete when every sufficiently large integer lies in $P(A)$. The Fibonacci numbers are $F_0=0$, $F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$.

Theorem (p. 2). Let $S=(s_1,s_2,\ldots)$ be the sequence of integers with $s_n=F_n-(-1)^n$ for $n\ge1$. Then $S$ has both properties stated on p. 1:

• (C) deleting any finite subsequence from $S$ leaves a complete sequence;
• (D) deleting any infinite subsequence from $S$ leaves a sequence that is not complete.

The first terms of $S$ are $2,0,3,2,6,7,14,20$, so $S$ is neither positive nor increasing at its start; the proof of (C) works with the tails $(s_k,s_{k+1},\ldots)$ for $k>4$ (p. 3).

Proof pointer

Pp. 2--10. For (D) (pp. 2--3): if the deleted terms are $s_{i_1},s_{i_2},\ldots$, the paper shows that $s_{i_n+1}-1$ is not in $P(S^*)$ for $n\ge4$, using the sum identity (1) on p. 3 to bound the sum of the remaining terms below $s_{i_n}$. For (C) (p. 3), it fixes $k>4$, writes $S'=(s_k,s_{k+1},\ldots)$, and defines "$P(S')$ has no gaps of length greater than $w$ beyond $x$" to mean that no $w+1$ consecutive integers above $x$ are missing from $P(S')$. Lemma 1 (p. 3, proved pp. 3--5) gives a finite $v$ with no gaps longer than $v$ beyond $s_k$; Lemma 2 (p. 3) lowers the gap bound from $w>0$ to $w-1$ beyond some term $s_i$, and is proved on pp. 5--10 from two auxiliary facts (a) and (b) (p. 5). Iterating Lemma 2 down to gap length $0$ shows that $S'$ is complete. The proof of (b) uses that the subset sums of a finite sequence are symmetric about half its total (p. 7).

Read depth

Claims checked: the definitions, the theorem and the statements of Lemmas 1 and 2 were read clause by clause on the page images of the print, and the proof of (D) was followed; the proof of (C) was read for structure. Nothing here is independently reviewed.

Dependencies

None in the corpus. The paper cites J. L. Brown, On complete sequences of integers, Amer. Math. Monthly 68 (1961), 557--560, for the completeness of the Fibonacci sequence and property (A).

Source. R. L. Graham, A property of Fibonacci numbers, Fibonacci Quart. 2 (1964), no. 1, 1--10; the edition read is named on the source card.

Bears on

• Problem 346: the theorem gives an explicit sequence with both deletion properties that the problem's hypothesis requires, complete after deleting any finite subsequence and not complete after deleting any infinite one. $S$ itself is not of the problem's form $1\le a_1<a_2<\cdots$, since $s_2=0$ and $s_4=s_1=2$, and the paper says nothing about its tails. The paper proves nothing about the ratios $s_{n+1}/s_n$ or their limit, which is what the problem asks about.