A297137 Numbers having an up-first zigzag pattern in base 7; see Comments.

9, 10, 11, 12, 13, 17, 18, 19, 20, 25, 26, 27, 33, 34, 41, 63, 64, 66, 67, 68, 69, 70, 71, 72, 74, 75, 76, 77, 78, 79, 80, 82, 83, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 119, 120, 121, 123, 124, 125, 126, 127, 128, 129, 131, 132, 133, 134, 135, 136

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 A297137-A297139 partition the natural numbers. See the guide at A297146.

Examples

			Base-7 digits of 4751: 1,6,5,6,5, with pattern UDUD, so that 4751 is in the sequence.
		

Links

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

Crossrefs

Cf. A297138, A297139.

Programs

Maple

read("transforms") :
isA297137 := proc(n)
    local dgs,ud;
    dgs := convert(n,base,7) ;
    if nops(dgs) < 2 then
        return false;
    end if;
    ud := DIFF(dgs) ;
    if 0 in ud then
        return false;
    else
        simplify( op(-1,ud) < 0) ;
    end if;
end proc:
for n from 1 to 200 do
    if isA297137(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jan 18 2018

Mathematica

a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
b = 7; t = Table[a[n, b], {n, 1, 10*z}];
u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297137 *)
v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297138 *)
Complement[Range[z], Union[u, v]]  (* A297139 *)