A297144 Numbers having a down-first zigzag pattern in base 9; see Comments.

9, 18, 19, 27, 28, 29, 36, 37, 38, 39, 45, 46, 47, 48, 49, 54, 55, 56, 57, 58, 59, 63, 64, 65, 66, 67, 68, 69, 72, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 163, 164, 165, 166, 167, 168, 169, 170, 171, 173, 174, 175, 176, 177, 178, 179, 244

Offset: 1

Author

Clark Kimberling, Jan 15 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 A297143-A297145 partition the natural numbers. See the guide at A297146.

Examples

			Base-9 digits of 7280: 1,0,8,7,8, with pattern DUDU, so that 7280 is in the sequence.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A297143, A297145.

Programs

Maple

filter:= proc(n)  local L;
L:= convert(n,base,9);
not has(L[2..-1]-L[1..-2],0) and L[-1]>L[-2]
end proc:
select(filter, [$9..1000]); # Robert Israel, Dec 06 2018

Mathematica

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 9; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297143 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297144 *)
Complement[Range[z], Union[u, v]]  (* A297145 *)