Theorem. Every positive integer can be represented as a weighted sum of Fibonacci squares.
Proof. Start with a postive integer $N$. Obtain its unique sum of non-consecutive Fibonacci numbers representation (i.e., the Zeckendorf representation). Now, rewrite the sum by replacing every Fibonacci number with Fibonacci squares using the following identities based on the index:
$$ \begin{align} F_{2n+1} & = F_n^2 + F_{n+1}^2 &; \text{ for odd indices } \\ F_{2n} & = F_n^2 + F_{n-1}^2 + \overbrace{F_{2(n-1)}}^\text{reduced even index} &; \text{ for even indices } \end{align} $$
We can recursively apply the identities to the reduction of $F_{2(n-1)}$ using the $F_{2n}$ identity until we are left with $F_1^2$ or $F_2^2$ when the recursion ends. For consistency, we can choose to represent $1$ as $F_1^2$.
This process results in a multiset of Fibonacci squares that can be summed to write $N$ as a weighted sum of Fibonacci squares.
$\blacksquare$
Question: What is the maximum weight possible in a sum of Fibonacci squares representation obtained using the process described above?
Empirical data. Data for $1 .. 20000$ shows the maximum weight is $16$. Obviously, the maximum weight keeps growing as $N$ grows.
I think the maximum weight grows as $O(\log_{\phi} N)$ where $$\phi = \frac{1+\sqrt{5}}{2}.$$
Is this correct?
$$ \begin{align} 1 &= 1 \times 1^2 \\ 2 &= 2 \times 1^2 \\ 3 &= 3 \times 1^2 \\ 4 &= 4 \times 1^2 \\ 5 &= 5 \times 1^2 \\ 6 &= 2 \times 1^2 + 1 \times 2^2 \\ 7 &= 3 \times 1^2 + 1 \times 2^2 \\ 8 &= 4 \times 1^2 + 1 \times 2^2 \\ 9 &= 1 \times 2^2 + 5 \times 1^2 \\ 10 &= 1 \times 2^2 + 6 \times 1^2 \\ 11 &= 1 \times 2^2 + 7 \times 1^2 \\ 12 &= 1 \times 2^2 + 8 \times 1^2 \\ 13 &= 2 \times 2^2 + 5 \times 1^2 \\ 14 &= 1 \times 2^2 + 1 \times 3^2 + 1 \times 1^2 \\ 15 &= 1 \times 2^2 + 1 \times 3^2 + 2 \times 1^2 \\ 16 &= 1 \times 2^2 + 1 \times 3^2 + 3 \times 1^2 \\ 17 &= 1 \times 2^2 + 1 \times 3^2 + 4 \times 1^2 \\ 18 &= 2 \times 2^2 + 1 \times 3^2 + 1 \times 1^2 \\ 19 &= 2 \times 2^2 + 1 \times 3^2 + 2 \times 1^2 \\ 20 &= 2 \times 2^2 + 1 \times 3^2 + 3 \times 1^2 \\ 21 &= 2 \times 2^2 + 1 \times 3^2 + 4 \times 1^2 \\ 22 &= 1 \times 3^2 + 2 \times 2^2 + 5 \times 1^2 \\ 23 &= 1 \times 3^2 + 2 \times 2^2 + 6 \times 1^2 \\ 24 &= 1 \times 3^2 + 2 \times 2^2 + 7 \times 1^2 \\ 25 &= 1 \times 3^2 + 2 \times 2^2 + 8 \times 1^2 \\ 26 &= 1 \times 3^2 + 3 \times 2^2 + 5 \times 1^2 \\ 27 &= 1 \times 3^2 + 3 \times 2^2 + 6 \times 1^2 \\ 28 &= 1 \times 3^2 + 3 \times 2^2 + 7 \times 1^2 \\ 29 &= 1 \times 3^2 + 3 \times 2^2 + 8 \times 1^2 \\ 30 &= 1 \times 3^2 + 3 \times 2^2 + 9 \times 1^2 \\ 31 &= 1 \times 3^2 + 3 \times 2^2 + 10 \times 1^2 \\ 32 &= 1 \times 3^2 + 3 \times 2^2 + 11 \times 1^2 \\ 33 &= 1 \times 3^2 + 3 \times 2^2 + 12 \times 1^2 \\ 34 &= 2 \times 3^2 + 3 \times 2^2 + 4 \times 1^2 \\ 35 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 1^2 \\ 36 &= 1 \times 3^2 + 1 \times 5^2 + 2 \times 1^2 \\ 37 &= 1 \times 3^2 + 1 \times 5^2 + 3 \times 1^2 \\ 38 &= 1 \times 3^2 + 1 \times 5^2 + 4 \times 1^2 \\ 39 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 1^2 + 1 \times 2^2 \\ 40 &= 1 \times 3^2 + 1 \times 5^2 + 2 \times 1^2 + 1 \times 2^2 \\ 41 &= 1 \times 3^2 + 1 \times 5^2 + 3 \times 1^2 + 1 \times 2^2 \\ 42 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 4 \times 1^2 \\ 43 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 5 \times 1^2 \\ 44 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 6 \times 1^2 \\ 45 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 7 \times 1^2 \\ 46 &= 1 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 8 \times 1^2 \\ 47 &= 2 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 \\ 48 &= 2 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 1 \times 1^2 \\ 49 &= 2 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 2 \times 1^2 \\ 50 &= 2 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 3 \times 1^2 \\ 51 &= 2 \times 3^2 + 1 \times 5^2 + 1 \times 2^2 + 4 \times 1^2 \\ 52 &= 2 \times 3^2 + 1 \times 5^2 + 2 \times 2^2 + 1 \times 1^2 \\ 53 &= 2 \times 3^2 + 1 \times 5^2 + 2 \times 2^2 + 2 \times 1^2 \\ 54 &= 2 \times 3^2 + 1 \times 5^2 + 2 \times 2^2 + 3 \times 1^2 \\ 55 &= 2 \times 3^2 + 1 \times 5^2 + 2 \times 2^2 + 4 \times 1^2 \\ 56 &= 1 \times 5^2 + 2 \times 3^2 + 2 \times 2^2 + 5 \times 1^2 \\ 57 &= 1 \times 5^2 + 2 \times 3^2 + 2 \times 2^2 + 6 \times 1^2 \\ 58 &= 1 \times 5^2 + 2 \times 3^2 + 2 \times 2^2 + 7 \times 1^2 \\ 59 &= 1 \times 5^2 + 2 \times 3^2 + 2 \times 2^2 + 8 \times 1^2 \\ 60 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 5 \times 1^2 \\ 61 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 6 \times 1^2 \\ 62 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 7 \times 1^2 \\ 63 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 8 \times 1^2 \\ 64 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 9 \times 1^2 \\ 65 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 10 \times 1^2 \\ 66 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 11 \times 1^2 \\ 67 &= 1 \times 5^2 + 2 \times 3^2 + 3 \times 2^2 + 12 \times 1^2 \\ 68 &= 1 \times 5^2 + 3 \times 3^2 + 3 \times 2^2 + 4 \times 1^2 \\ 69 &= 1 \times 5^2 + 3 \times 3^2 + 3 \times 2^2 + 5 \times 1^2 \\ 70 &= 1 \times 5^2 + 3 \times 3^2 + 3 \times 2^2 + 6 \times 1^2 \\ 71 &= 1 \times 5^2 + 3 \times 3^2 + 3 \times 2^2 + 7 \times 1^2 \\ 72 &= 1 \times 5^2 + 3 \times 3^2 + 3 \times 2^2 + 8 \times 1^2 \\ 73 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 5 \times 1^2 \\ 74 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 6 \times 1^2 \\ 75 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 7 \times 1^2 \\ 76 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 8 \times 1^2 \\ 77 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 9 \times 1^2 \\ 78 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 10 \times 1^2 \\ 79 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 11 \times 1^2 \\ 80 &= 1 \times 5^2 + 3 \times 3^2 + 4 \times 2^2 + 12 \times 1^2 \\ 81 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 9 \times 1^2 \\ 82 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 10 \times 1^2 \\ 83 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 11 \times 1^2 \\ 84 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 12 \times 1^2 \\ 85 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 13 \times 1^2 \\ 86 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 14 \times 1^2 \\ 87 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 15 \times 1^2 \\ 88 &= 1 \times 5^2 + 3 \times 3^2 + 5 \times 2^2 + 16 \times 1^2 \\ 89 &= 2 \times 5^2 + 3 \times 3^2 + 2 \times 2^2 + 4 \times 1^2 \\ 90 &= 1 \times 5^2 + 1 \times 8^2 + 1 \times 1^2 \\ 91 &= 1 \times 5^2 + 1 \times 8^2 + 2 \times 1^2 \\ 92 &= 1 \times 5^2 + 1 \times 8^2 + 3 \times 1^2 \\ 93 &= 1 \times 5^2 + 1 \times 8^2 + 4 \times 1^2 \\ 94 &= 1 \times 5^2 + 1 \times 8^2 + 1 \times 1^2 + 1 \times 2^2 \\ 95 &= 1 \times 5^2 + 1 \times 8^2 + 2 \times 1^2 + 1 \times 2^2 \\ 96 &= 1 \times 5^2 + 1 \times 8^2 + 3 \times 1^2 + 1 \times 2^2 \\ 97 &= 1 \times 5^2 + 1 \times 8^2 + 1 \times 2^2 + 4 \times 1^2 \\ 98 &= 1 \times 5^2 + 1 \times 8^2 + 1 \times 2^2 + 5 \times 1^2 \\ 99 &= 1 \times 5^2 + 1 \times 8^2 + 1 \times 2^2 + 6 \times 1^2 \\ 100 &= 1 \times 5^2 + 1 \times 8^2 + 1 \times 2^2 + 7 \times 1^2 \\ \cdots \\ 19997 &= 1 \times 89^2 + 2 \times 55^2 + 3 \times 34^2 + 4 \times 21^2 + 3 \times 13^2 + 2 \times 8^2 + 3 \times 5^2 + 5 \times 3^2 + 6 \times 2^2 + 15 \times 1^2 \\ 19998 &= 1 \times 89^2 + 2 \times 55^2 + 3 \times 34^2 + 4 \times 21^2 + 3 \times 13^2 + 2 \times 8^2 + 3 \times 5^2 + 5 \times 3^2 + 6 \times 2^2 + 16 \times 1^2 \\ 19999 &= 1 \times 89^2 + 2 \times 55^2 + 3 \times 34^2 + 4 \times 21^2 + 3 \times 13^2 + 2 \times 8^2 + 3 \times 5^2 + 5 \times 3^2 + 7 \times 2^2 + 13 \times 1^2 \\ 20000 &= 1 \times 89^2 + 2 \times 55^2 + 3 \times 34^2 + 4 \times 21^2 + 3 \times 13^2 + 2 \times 8^2 + 3 \times 5^2 + 5 \times 3^2 + 7 \times 2^2 + 14 \times 1^2 \\ \cdots \end{align} $$
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This line of inquiry came up while thinking about this question.