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A297140 Numbers having an up-first zigzag pattern in base 8; see Comments.

%I A297140 #4 Jan 15 2018 08:55:31
%S A297140 10,11,12,13,14,15,19,20,21,22,23,28,29,30,31,37,38,39,46,47,55,80,81,
%T A297140 83,84,85,86,87,88,89,90,92,93,94,95,96,97,98,99,101,102,103,104,105,
%U A297140 106,107,108,110,111,112,113,114,115,116,117,119,120,121,122
%N A297140 Numbers having an up-first zigzag pattern in base 8; see Comments.
%C A297140 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 A297140 Base-8 digits of 3575: 6, 7, 6, 7, with pattern UDU, so that 3575 is in the sequence.
%t A297140 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297140 b = 8; t = Table[a[n, b], {n, 1, 10*z}];
%t A297140 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297140 *)
%t A297140 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297141 *)
%t A297140 Complement[Range[z], Union[u, v]]  (* A297142 *)
%Y A297140 Cf. A297141, A297142.
%K A297140 nonn,easy,base
%O A297140 1,1
%A A297140 _Clark Kimberling_, Jan 15 2018