cp's OEIS Frontend

A297129 Numbers having a down-first zigzag pattern in base 4; see Comments.

%I A297129 #6 Jan 14 2018 23:01:14
%S A297129 4,8,9,12,13,14,17,18,19,33,34,35,36,38,39,49,50,51,52,54,55,56,57,59,
%T A297129 68,70,71,72,73,75,76,77,78,132,134,135,136,137,139,140,141,142,145,
%U A297129 146,147,152,153,155,156,157,158,196,198,199,200,201,203,204,205
%N A297129 Numbers having a down-first zigzag pattern in base 4; see Comments.
%C A297129 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 A297129 Base-4 digits of 5000: 1,0,3,2,0,2,0, with pattern DUDUD, so that 5000 is in the sequence.
%t A297129 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297129 b = 4; t = Table[a[n, b], {n, 1, 10*z}];
%t A297129 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297128 *)
%t A297129 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297129 *)
%t A297129 Complement[Range[z], Union[u, v]]  (* A297130 *)
%Y A297129 Cf. A297128, A297130.
%K A297129 nonn,easy,base
%O A297129 1,1
%A A297129 _Clark Kimberling_, Jan 14 2018