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Base-5 variant of Knuth's A220952. The definition of the sequence by an adjacency diagram is the same as in A220952, except that the diagram for the sequence here is:

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(0,4)--(1,4)--(2,4)--(3,4) (4,4)

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(0,3) (1,3)--(2,3)--(3,3) (4,3)

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(0,2) (1,2)--(2,2)--(3,2) (4,2)

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(0,1) (1,1)--(2,1)--(3,1) (4,1)

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(0,0) (1,0)--(2,0)--(3,0)--(4,0)

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Conjecture: As in A220952, it can be proved (a) that every positive integer is adjacent to exactly two nonnegative integers, and (b) that with this definition of adjacency, the nonnegative integers form a path starting with 0.

The adjacency definition implies that the terms, when written with 3 base-5 digits, define the coordinates of a self-avoiding, space-filling path in a 5 X 5 X 5 cube. All 3 orthogonal projections to the plane are congruent to the diagram above. This property is maintained in the 4th, 5th ... dimension.

The variants of such adjacency diagrams may be distinguished by letter codes, in this case "Hs" with "H" for the vertical bars (0,0..4), (4,0..4), and "s" for the inner structure (1..3,1..3). Knuth's A220952 would then be denoted by "Hn".

Base-5 variant of Knuth's A220952. The definition of the sequence by an adjacency diagram is the same as in A220952, except that the diagram for the sequence here is:

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(0,4)--(1,4) (2,4)--(3,4) (4,4)

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(0,3) (1,3) (2,3) (3,3) (4,3)

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(0,2) (1,2) (2,2) (3,2)--(4,2)

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(0,1) (1,1) (2,1)--(3,1)--(4,1)

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(0,0) (1,0)--(2,0)--(3,0)--(4,0)

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Conjecture: As in A220952, it can be proved (a) that every positive integer is adjacent to exactly two nonnegative integers, and (b) that with this definition of adjacency, the nonnegative integers form a path starting with 0.

The adjacency definition implies that the terms, when written with 3 base-5 digits, define the coordinates of a self-avoiding, space-filling path in a 5 X 5 X 5 cube. All 3 orthogonal projections to the plane are congruent to the diagram above. This property is maintained in the 4th, 5th ... dimension.

The variants of such adjacency diagrams may be distinguished by letter codes, in this case "Ln" with "L" for the path (0,0)...(2,1), and "n" for the path in the upper right corner which has the same shape as the inner structure (1,1)...(3,3) of Knuth's A220952.

The concept of the adjacency diagram in A220952 is not limited to base 5, but can be investigated for all odd bases. In "base 1", the path consists of a single node (1,1). For base 3, there is the ternary Gray code A128173 only. A corresponding simple up-down pattern with a path (0,0), ... (0,n), (1,n), ... (1,0), (2,0), ... (n,n) can be constructed for any odd base.

Conjecture: a(4) = 673.