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

%I A297137 #11 Jan 18 2018 16:26:19
%S A297137 9,10,11,12,13,17,18,19,20,25,26,27,33,34,41,63,64,66,67,68,69,70,71,
%T A297137 72,74,75,76,77,78,79,80,82,83,84,85,86,87,88,90,91,92,93,94,95,96,
%U A297137 119,120,121,123,124,125,126,127,128,129,131,132,133,134,135,136
%N A297137 Numbers having an up-first zigzag pattern in base 7; see Comments.
%C A297137 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.
%H A297137 Robert Israel, <a href="/A297137/b297137.txt">Table of n, a(n) for n = 1..10000</a>
%e A297137 Base-7 digits of 4751: 1,6,5,6,5, with pattern UDUD, so that 4751 is in the sequence.
%p A297137 read("transforms") :
%p A297137 isA297137 := proc(n)
%p A297137     local dgs,ud;
%p A297137     dgs := convert(n,base,7) ;
%p A297137     if nops(dgs) < 2 then
%p A297137         return false;
%p A297137     end if;
%p A297137     ud := DIFF(dgs) ;
%p A297137     if 0 in ud then
%p A297137         return false;
%p A297137     else
%p A297137         simplify( op(-1,ud) < 0) ;
%p A297137     end if;
%p A297137 end proc:
%p A297137 for n from 1 to 200 do
%p A297137     if isA297137(n) then
%p A297137         printf("%d,",n) ;
%p A297137     end if;
%p A297137 end do: # _R. J. Mathar_, Jan 18 2018
%t A297137 a[n_, b_] := Sign[Differences[IntegerDigits[n, b]]]; z = 300;
%t A297137 b = 7; t = Table[a[n, b], {n, 1, 10*z}];
%t A297137 u = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == 1 &]   (* A297137 *)
%t A297137 v = Select[Range[z], ! MemberQ[t[[#]], 0] && First[t[[#]]] == -1 &]  (* A297138 *)
%t A297137 Complement[Range[z], Union[u, v]]  (* A297139 *)
%Y A297137 Cf. A297138, A297139.
%K A297137 nonn,easy,base
%O A297137 1,1
%A A297137 _Clark Kimberling_, Jan 15 2018