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

1,3

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.

Links

Table of n, a(n) for n=1..61.

Example

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.

Crossrefs

Sequence in context: A171886 A343238 A018559 * A057196 A080637 A124134

Adjacent sequences: A384311 A384312 A384313 * A384315 A384316 A384317

Keyword

base,easy,nonn

Author

Frederik P.J. Vandecasteele, May 25 2025

Status

approved