State/Values in the range topograph adjacent to a given face form a quadratic sequence

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Proposition
Proof

Proposition: (Values in the range topograph adjacent to a given face form a quadratic sequence) Suppose that $Q$ is a integer-valued binary quadratic form. Suppose that $p$ is the value on a face of the range topograph of $Q$. Let $(\ldots, n_{-2}, n_{-1}, n_0, n_1, n_2, \ldots)$ denote the sequence of values on faces adjacent to $p$.

Then, the sequence $(n_i)$ (for $i \in \ZZ$) is a quadratic sequence of acceleration $p$.

Logical Connections

This statement logically relies on the following definitions and statements: Def/Binary quadratic form, Def/Range topograph, Def/Quadratic sequence, State/Arithmetic progression rule for binary quadratic forms, Def/Sequence of differences

The following statements and definitions rely on the material of this page: State/Every arc of river is finite

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State/Values in the range topograph adjacent to a given face form a quadratic sequence

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	Suppose that $Q$ is a binary quadratic form, $p$ is the value on a face of the range topograph of $Q$, and let $(\ldots, n_{-2}, n_{-1}, n_0, n_1, n_2, \ldots)$ denote the sequence of values on faces adjacent to $p$.
	By the arithmetic progression rule for binary quadratic forms, we find that: $n_{i}, n_{i+1} + p, n_{i+2}$ is an arithmetic progression for all $i \in \ZZ$. It follows algebraically that: $n_{i+2} = - n_i + 2(n_{i+1} + p)$. Consider the sequence of differences obtained from the sequence $n_i$. Namely, consider the sequence $(n_i')$ (for all $i \in \ZZ$), given by: $$n_i' = n_{i+1} - n_i.$$ Consider the sequence of differences obtained from the sequence $n_i'$, i.e., the second-difference sequence of $(n_i)$ : $$n_i' ' = n_{i+1}' - n_i' = n_{i+2} - 2 n_{i+1} + n_i.$$ Then, the arithmetic progression considered above yields: $n_i' ' = n_{i+2} - 2 n_{i+1} + n_i = (-n_i + 2(n_{i+1} + p)) - 2 n_{i+1} + n_i = 2p$. It follows that the sequence of second-differences $(n_i' ')$ is a constant sequence, with constant $2p$.
	Hence the sequence $(n_i)$ is a quadratic sequence of acceleration $p$.