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A297128 Numbers having an up-first zigzag pattern in base 4; see Comments.

%I A297128 #7 Jan 14 2018 18:23:42
%S A297128 6,7,11,24,25,27,28,29,30,44,45,46,97,98,99,100,102,103,108,109,110,
%T A297128 113,114,115,116,118,119,120,121,123,177,178,179,180,182,183,184,185,
%U A297128 187,388,390,391,392,393,395,396,397,398,401,402,403,408,409,411,412
%N A297128 Numbers having an up-first zigzag pattern in base 4; see Comments.
%C A297128 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 A297128..A297130 partition the natural numbers.  See the guide at A297146.
%e A297128 Base-4 digits of 3003: 2,3,2,3,2,3, with pattern UDUDU, so that 3003 is in the sequence.
%t A297128 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297128 b = 4; t = Table[a[n, b], {n, 1, 10*z}];
%t A297128 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297128 *)
%t A297128 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297129 *)
%t A297128 Complement[Range[z], Union[u, v]]  (* A297130 *)
%Y A297128 Cf. A297129, A297130.
%K A297128 nonn,easy,base
%O A297128 1,1
%A A297128 _Clark Kimberling_, Jan 13 2018