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A297147 Numbers having a down-first zigzag pattern in base 10; see Comments.

%I A297147 #6 Apr 11 2018 09:03:22
%S A297147 10,20,21,30,31,32,40,41,42,43,50,51,52,53,54,60,61,62,63,64,65,70,71,
%T A297147 72,73,74,75,76,80,81,82,83,84,85,86,87,90,91,92,93,94,95,96,97,98,
%U A297147 101,102,103,104,105,106,107,108,109,201,202,203,204,205,206,207
%N A297147 Numbers having a down-first zigzag pattern in base 10; see Comments.
%C A297147 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 A297146-A297148 partition the natural numbers.  See the guide at A297146.
%e A297147 Base-10 digits of 65498: 6,5,4,9,8, with pattern DUD, so that 65498 is in the sequence.
%t A297147 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297147 b = 10; t = Table[a[n, b], {n, 1, 10*z}];
%t A297147 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297146 *)
%t A297147 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297147 *)
%t A297147 Complement[Range[z], Union[u, v]]  (* A297148 *)
%Y A297147 Cf. A297146, A297148.
%K A297147 nonn,easy,base
%O A297147 1,1
%A A297147 _Clark Kimberling_, Jan 15 2018