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