%I
%S 8,16,17,24,25,26,32,33,34,35,40,41,42,43,44,48,49,50,51,52,53,56,57,
%T 58,59,60,61,62,65,66,67,68,69,70,71,129,130,131,132,133,134,135,136,
%U 138,139,140,141,142,143,193,194,195,196,197,198,199,200,202,203
%N Numbers having a downfirst zigzag pattern in base 8; see Comments.
%C A number n having baseb digits d(m), d(m1),..., 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 "upfirst zigzag pattern in base b"; in the latter, a "downfirst 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 upfirst zigzag pattern in base 10, whereas 4,2,5,3,0,1,4,2 has a downfirst pattern. The sequences A297140A297142 partition the natural numbers. See the guide at A297146.
%e Base8 digits of 4599: 1,0,7,6,7, with pattern DUDU, so that 4599 is in the sequence.
%t a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t b = 8; t = Table[a[n, b], {n, 1, 10*z}];
%t u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297140 *)
%t v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297141 *)
%t Complement[Range[z], Union[u, v]] (* A297142 *)
%Y Cf. A297140, A297142.
%K nonn,easy,base
%O 1,1
%A _Clark Kimberling_, Jan 15 2018
