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A297135 Numbers having a down-first zigzag pattern in base 6; see Comments.

%I A297135 #6 Jan 14 2018 23:02:09
%S A297135 6,12,13,18,19,20,24,25,26,27,30,31,32,33,34,37,38,39,40,41,73,74,75,
%T A297135 76,77,78,80,81,82,83,109,110,111,112,113,114,116,117,118,119,120,121,
%U A297135 123,124,125,145,146,147,148,149,150,152,153,154,155,156,157,159
%N A297135 Numbers having a down-first zigzag pattern in base 6; see Comments.
%C A297135 A number n having base-b digits d(m), d(m-1),..., d(0) such that d(i) != d(i+1) for 0 <= i < m shows a zigzag pattern of one or more segments, in the following sense.  Writing U for up and D for down, there are two kinds of patterns:  U, UD, UDU, UDUD, ... and D, DU, DUD, DUDU, ... .  In the former case, we say n has an "up-first zigzag pattern in base b"; in the latter, a "down-first zigzag pattern in base b".  Example:  2,4,5,3,0,1,4,2 has segments 2,4,5; 5,3,0; 0,1,4; and 4,2, so that 24530142, with pattern UDUD, has an up-first zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a down-first pattern.  The sequences A297134-A297136 partition the natural numbers.  See the guide at A297146.
%e A297135 Base-6 digits of 4529: 3,2,5,4,5, with pattern DUDU, so that 4529 is in the sequence.
%t A297135 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297135 b = 6; t = Table[a[n, b], {n, 1, 10*z}];
%t A297135 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297134 *)
%t A297135 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297135 *)
%t A297135 Complement[Range[z], Union[u, v]]  (* A297136 *)
%Y A297135 Cf. A297134, A297136.
%K A297135 nonn,easy,base
%O A297135 1,1
%A A297135 _Clark Kimberling_, Jan 14 2018