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%I #9 Dec 06 2018 16:35:40
%S 9,18,19,27,28,29,36,37,38,39,45,46,47,48,49,54,55,56,57,58,59,63,64,
%T 65,66,67,68,69,72,73,74,75,76,77,78,79,82,83,84,85,86,87,88,89,163,
%U 164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,244
%N Numbers having a down-first zigzag pattern in base 9; 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 A297143-A297145 partition the natural numbers. See the guide at A297146.
%H Robert Israel, <a href="/A297144/b297144.txt">Table of n, a(n) for n = 1..10000</a>
%e Base-9 digits of 7280: 1,0,8,7,8, with pattern DUDU, so that 7280 is in the sequence.
%p filter:= proc(n) local L;
%p L:= convert(n,base,9);
%p not has(L[2..-1]-L[1..-2],0) and L[-1]>L[-2]
%p end proc:
%p select(filter, [$9..1000]); # _Robert Israel_, Dec 06 2018
%t a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t b = 9; t = Table[a[n, b], {n, 1, 10*z}];
%t u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297143 *)
%t v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &] (* A297144 *)
%t Complement[Range[z], Union[u, v]] (* A297145 *)
%Y Cf. A297143, A297145.
%K nonn,easy,base
%O 1,1
%A _Clark Kimberling_, Jan 15 2018
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