OFFSET

1,1

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 A297131-A297133 partition the natural numbers. See the guide at A297146.

EXAMPLE

Base-5 digits of 4973: 1,2,4,3,4,3, with pattern UDUD, so that 4973 is in the sequence.

MATHEMATICA

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;

b = 5; t = Table[a[n, b], {n, 1, 10*z}];

u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &] (* A297131 *)

v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &] (* A297132 *)

Complement[Range[z], Union[u, v]] (* A297133 *)

CROSSREFS

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Jan 14 2018

STATUS

approved