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A297141 Numbers having a down-first zigzag pattern in base 8; see Comments.

%I A297141 #7 Apr 11 2018 09:03:03
%S A297141 8,16,17,24,25,26,32,33,34,35,40,41,42,43,44,48,49,50,51,52,53,56,57,
%T A297141 58,59,60,61,62,65,66,67,68,69,70,71,129,130,131,132,133,134,135,136,
%U A297141 138,139,140,141,142,143,193,194,195,196,197,198,199,200,202,203
%N A297141 Numbers having a down-first zigzag pattern in base 8; see Comments.
%C A297141 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 A297140-A297142 partition the natural numbers.  See the guide at A297146.
%e A297141 Base-8 digits of 4599: 1,0,7,6,7, with pattern DUDU, so that 4599 is in the sequence.
%t A297141 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297141 b = 8; t = Table[a[n, b], {n, 1, 10*z}];
%t A297141 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297140 *)
%t A297141 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297141 *)
%t A297141 Complement[Range[z], Union[u, v]]  (* A297142 *)
%Y A297141 Cf. A297140, A297142.
%K A297141 nonn,easy,base
%O A297141 1,1
%A A297141 _Clark Kimberling_, Jan 15 2018