Complete paths in a linear lattice

A brief treatment of complete paths within a bounded linear lattice is presented here for completeness.

Clearly, the terms non-self-adjacent and non-self-intersecting are irrelevant in this case.

A linear lattice is bounded by two nodes.

Complete paths start at any node within the lattice including the a boundary node and end with one of the boundary nodes.

The shape of a path is defined entirely by its length.

Let the number of nodes in the bounded lattice be n, with n > 1.

If the start node is a boundary node then there is one complete path of length n.

If the start node is an intermediate node, say the mth, then there is one complete path of length m and a second of length n - m + 1.

Consequently, there are 2 complete paths of each length in the range 2 <= path length <= n.

Over all starting nodes, the total number of complete paths is 2(n - 1).

Each boundary node will be the end node for (n - 1) paths.

Each node will appear in (n - 1) paths.

Node Characteristics

No. of neighbors	No. of nodes		Description
	1			2		boundary nodes
	2			n - 2		intermediate nodes
Total				n

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