A384314 Numbers k such that the nonzero digits in the ternary expansion k = d(1),...,d(m) satisfy d(2*i+1) = d(1) and d(2*i) = 3-d(1).

0, 1, 2, 3, 5, 6, 7, 9, 10, 15, 16, 18, 20, 21, 23, 27, 29, 30, 32, 45, 47, 48, 50, 54, 55, 60, 61, 63, 64, 69, 70, 81, 82, 87, 88, 90, 91, 96, 97, 135, 136, 141, 142, 144, 145, 150, 151, 162, 164, 165, 167, 180, 182, 183, 185, 189, 191, 192, 194, 207, 209

Offset: 1

Author

Frederik P.J. Vandecasteele, May 25 2025

Keywords

• base
• easy
• nonn

Comments

The ternary expansion of the numbers in this sequence correspond to a valid linear gear train configurations with pairwise intermeshing of neighboring gears: 0 for an idler gear, 1 for a gear driven in rotational direction A, 2 for a gear driven in rotational direction B.

A gear train is valid if it has no contradictions, where a contradiction occurs if two meshed gears rotate in the same direction.

The rotation directions for the whole train are determined by the most significant ternary digit 1 or 2.

Any later driven gears must be in the same direction as the most significant when at an even distance away from there and the opposite direction when an odd distance away.

Examples

			32 is a term since its ternary expansion is
   ternary     1 0 1 2
   direction   A B A B
The direction for the 0 gear is determined by its preceding A, and the whole train has valid alternating A,B adjacent pairs.
11 is not a term because its ternary expansion 102 does not follow the pattern ABA.