%I #7 Apr 11 2018 09:02:49
%S 7,14,15,21,22,23,28,29,30,31,35,36,37,38,39,42,43,44,45,46,47,50,51,
%T 52,53,54,55,99,100,101,102,103,104,105,107,108,109,110,111,148,149,
%U 150,151,152,153,154,156,157,158,159,160,161,162,164,165,166,167
%N Numbers having a down-first zigzag pattern in base 7; 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 A297137-A297139 partition the natural numbers. See the guide at A297146.
%e Base-7 digits of 5000: 2,0,4,0,2, with pattern DUDU, so that 5000 is in the sequence.
%t a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t b = 7; t = Table[a[n, b], {n, 1, 10*z}];
%t u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297137 *)
%t v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &] (* A297138 *)
%t Complement[Range[z], Union[u, v]] (* A297139 *)
%Y Cf. A297137, A297139.
%K nonn,easy,base
%O 1,1
%A _Clark Kimberling_, Jan 15 2018