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%I #6 Jan 14 2018 23:02:01
%S 8,9,10,11,15,16,17,22,23,29,48,49,51,52,53,54,55,56,58,59,60,61,62,
%T 63,65,66,67,68,69,70,90,91,92,94,95,96,97,98,99,101,102,103,104,105,
%U 106,132,133,134,135,137,138,139,140,141,142,174,175,176,177,178
%N Numbers having an up-first zigzag pattern in base 6; see Comments.
%C 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 A297134-A297136 partition the natural numbers. See the guide at A297146.
%e Base-6 digits of 5000: 3,5,0,5,2, with pattern UDUD, so that 5000 is in the sequence.
%t a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t b = 6; t = Table[a[n, b], {n, 1, 10*z}];
%t u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297134 *)
%t v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &] (* A297135 *)
%t Complement[Range[z], Union[u, v]] (* A297136 *)
%Y Cf. A297135, A297136.
%K nonn,easy,base
%O 1,1
%A _Clark Kimberling_, Jan 14 2018
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