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

1,1

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

Links

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

Example

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

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 *)

Crossrefs

Cf. A297138, A297139.

Sequence in context: A268360 A297263 A296704 * A129849 A274562 A258469

Adjacent sequences: A297134 A297135 A297136 * A297138 A297139 A297140

Keyword

nonn,easy,base

Author

Clark Kimberling, Jan 15 2018

Status

approved