A307403 Base-5 based twisted permutation of the nonnegative integers - variant "Hs".

0, 1, 2, 3, 4, 9, 14, 19, 18, 13, 8, 7, 12, 17, 16, 11, 6, 5, 10, 15, 20, 21, 22, 23, 24, 49, 74, 99, 98, 73, 48, 47, 72, 97, 96, 71, 46, 45, 70, 95, 90, 65, 40, 35, 60, 85, 80, 55, 30, 31, 56, 81, 86, 61, 36, 41, 66, 91, 92, 67, 42, 37, 62

Offset: 0

Author

Georg Fischer, Apr 07 2019

Keywords

• nonn
• base
• easy

Comments

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".

Examples

			In base-5, the terms for the path in two dimensions are 0, 1, 2, 3, 4, 14, 24, 34, 33, 23, 13, 12, 22, 32, 31, 21, 11, 10, 20, 30, 40, 41, 42, 43, 44.
		

Links

• Georg Fischer, Repository of programs for related sequences, (gen_paths.pl)

Crossrefs

Cf. A220952, (main entry, "Hn"), A307404 ("Ln"), A307405 ("Ls"), A307406 (number of variants per odd base).

Programs

Perl

cf. link.