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A297146 Numbers having an up-first zigzag pattern in base 10; see Comments.

%I A297146 #6 Jan 15 2018 15:31:05
%S A297146 12,13,14,15,16,17,18,19,23,24,25,26,27,28,29,34,35,36,37,38,39,45,46,
%T A297146 47,48,49,56,57,58,59,67,68,69,78,79,89,120,121,123,124,125,126,127,
%U A297146 128,129,130,131,132,134,135,136,137,138,139,140,141,142,143,145
%N A297146 Numbers having an up-first zigzag pattern in base 10; see Comments.
%C A297146 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. In the following guide, column four, "complement" means the sequence of natural numbers not in the corresponding sequences in columns 2 and 3.
%C A297146 ***
%C A297146 Base      up-first    down-first  complement
%C A297146 2           (none)     A000975     A107907
%C A297146 3          A297124     A297125     A297126
%C A297146 4          A297128     A297129     A297130
%C A297146 5          A297131     A297132     A297133
%C A297146 6          A297134     A297135     A297136
%C A297146 7          A297137     A297138     A297139
%C A297146 8          A297140     A297141     A297142
%C A297146 9          A297143     A297144     A297145
%C A297146 10         A297146     A297147     A297148
%e A297146 Base-10 digits of 59898: 5,9,8,9,8, with pattern UDUD, so that 59898 is in the sequence.
%t A297146 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297146 b = 10; t = Table[a[n, b], {n, 1, 10*z}];
%t A297146 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297146 *)
%t A297146 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297147 *)
%t A297146 Complement[Range[z], Union[u, v]]  (* A297148 *)
%Y A297146 Cf. A297147, A297148.
%K A297146 nonn,easy,base
%O A297146 1,1
%A A297146 _Clark Kimberling_, Jan 15 2018