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Represent the doubly infinite ladder graph by $\mathbb{Z}\times \{0,1\}$, with horizontal edges between successive columns and a vertical rung at each column. An allowed self-avoiding walk is coded by a word in $R,V$. Two successive $V$ steps revisit the preceding graph vertex and are forbidden. Conversely, every word without consecutive $V$ steps is a self-avoiding walk: horizontal steps strictly increase the column, and a column is visited vertically at most once. This establishes a bijection, not merely an upper bound.

For the directed ladder self-avoiding walk count, ${\sigma }_0=1$, ${\sigma }_1=2$, and for $n\ge 2$ a word either ends in $R$, or ends in $RV$. Removing that final block givesLet $\phi =(1+\sqrt{5})/2$ be the golden ratio, and $\psi =(1-\sqrt{5})/2$. The Binet formula for the Fibonacci number yieldsSince $|\psi |<\phi$, taking $n$th roots gives ${\lim }_{n\to \infty }{\sigma }_n^{1/n}=\phi$. This use of the golden-ratio symbol is independent of the connection rate in Question 1.

True. Write the odd integer as $m=2n+1$. The identity from part (b) and $p\mid F_m$ giveConsecutive Fibonacci numbers are coprime, so $F_{n+1}$ is invertible modulo $p$. HenceThus $-1$ is a quadratic residue modulo the odd prime $p$, which occurs exactly when $\boxed{p\equiv 1(\mathrm{mod}4)}$.

The stated formula is Cassini's identity. It holds for $n=1$, and the recurrence giveswhich is the negative of the expression at index $n$. This proves the identity by induction.

ConsequentlyThe positive Fibonacci numbers tend to infinity, so this difference tends to zero.

The Fibonacci recurrence givesThe map $x\mapsto 1+1/x$ is strictly decreasing for $x>0$. Since $a_3=2\ge a_1=1$, applying this decreasing map reverses each inequality and proves by mathematical induction thatTaking even $n$ shows $a_{2n+2}\le a_{2n}$, so $(a_{2n})$ is a decreasing sequence.

PutThe initial values are $S_1=S_2=1$. Using Pascal's identity and the convention that an out-of-range binomial coefficient is zero,Thus $(S_n)$ has the same initial values and linear recurrence relation as the Fibonacci numbers. Mathematical induction gives