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A297132 Numbers having a down-first zigzag pattern in base 5; see Comments.

%I A297132 #6 Jan 14 2018 23:01:44
%S A297132 5,10,11,15,16,17,20,21,22,23,26,27,28,29,51,52,53,54,55,57,58,59,76,
%T A297132 77,78,79,80,82,83,84,85,86,88,89,101,102,103,104,105,107,108,109,110,
%U A297132 111,113,114,115,116,117,119,130,132,133,134,135,136,138,139,140
%N A297132 Numbers having a down-first zigzag pattern in base 5; see Comments.
%C A297132 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 A297131-A297133 partition the natural numbers.  See the guide at A297146.
%e A297132 Base-5 digits of 3723: 1,0,4,3,4,3, with pattern DUDUD, so that 3723 is in the sequence.
%t A297132 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297132 b = 5; t = Table[a[n, b], {n, 1, 10*z}];
%t A297132 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297131 *)
%t A297132 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297132 *)
%t A297132 Complement[Range[z], Union[u, v]]  (* A297133 *)
%Y A297132 Cf. A297131, A297133.
%K A297132 nonn,easy,base
%O A297132 1,1
%A A297132 _Clark Kimberling_, Jan 14 2018