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%I #6 Jan 15 2018 15:31:05
%S 12,13,14,15,16,17,18,19,23,24,25,26,27,28,29,34,35,36,37,38,39,45,46,
%T 47,48,49,56,57,58,59,67,68,69,78,79,89,120,121,123,124,125,126,127,
%U 128,129,130,131,132,134,135,136,137,138,139,140,141,142,143,145
%N Numbers having an up-first zigzag pattern in base 10; 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 A297146-A297148 partition the natural numbers. In the following guide, column four, "complement" means the sequence of natural numbers not in the corresponding sequences in columns 2 and 3.
%C ***
%C Base up-first down-first complement
%C 2 (none) A000975 A107907
%C 3 A297124 A297125 A297126
%C 4 A297128 A297129 A297130
%C 5 A297131 A297132 A297133
%C 6 A297134 A297135 A297136
%C 7 A297137 A297138 A297139
%C 8 A297140 A297141 A297142
%C 9 A297143 A297144 A297145
%C 10 A297146 A297147 A297148
%e Base-10 digits of 59898: 5,9,8,9,8, with pattern UDUD, so that 59898 is in the sequence.
%t a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t b = 10; t = Table[a[n, b], {n, 1, 10*z}];
%t u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297146 *)
%t v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &] (* A297147 *)
%t Complement[Range[z], Union[u, v]] (* A297148 *)
%Y Cf. A297147, A297148.
%K nonn,easy,base
%O 1,1
%A _Clark Kimberling_, Jan 15 2018
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