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

4, 8, 9, 12, 13, 14, 17, 18, 19, 33, 34, 35, 36, 38, 39, 49, 50, 51, 52, 54, 55, 56, 57, 59, 68, 70, 71, 72, 73, 75, 76, 77, 78, 132, 134, 135, 136, 137, 139, 140, 141, 142, 145, 146, 147, 152, 153, 155, 156, 157, 158, 196, 198, 199, 200, 201, 203, 204, 205

Offset: 1

Author

Clark Kimberling, Jan 14 2018

Keywords

• nonn
• easy
• base

Comments

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.

Examples

			Base-4 digits of 5000: 1,0,3,2,0,2,0, with pattern DUDUD, so that 5000 is in the sequence.
		

Crossrefs

Cf. A297128, A297130.

Programs

Mathematica

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 4; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297128 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297129 *)
Complement[Range[z], Union[u, v]]  (* A297130 *)