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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).

%I A384314 #29 Jun 18 2025 19:02:57
%S A384314 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,
%T A384314 60,61,63,64,69,70,81,82,87,88,90,91,96,97,135,136,141,142,144,145,
%U A384314 150,151,162,164,165,167,180,182,183,185,189,191,192,194,207,209
%N 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).
%C A384314 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.
%C A384314 A gear train is valid if it has no contradictions, where a contradiction occurs if two meshed gears rotate in the same direction.
%C A384314 The rotation directions for the whole train are determined by the most significant ternary digit 1 or 2.
%C A384314 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.
%e A384314 32 is a term since its ternary expansion is
%e A384314    ternary     1 0 1 2
%e A384314    direction   A B A B
%e A384314 The direction for the 0 gear is determined by its preceding A, and the whole train has valid alternating A,B adjacent pairs.
%e A384314 11 is not a term because its ternary expansion 102 does not follow the pattern ABA.
%K A384314 base,easy,nonn
%O A384314 1,3
%A A384314 _Frederik P.J. Vandecasteele_, May 25 2025