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A297268 Numbers whose base-9 digits have equal down-variation and up-variation; see Comments.

%I A297268 #8 Jan 23 2018 19:29:34
%S A297268 1,2,3,4,5,6,7,8,10,20,30,40,50,60,70,80,82,91,100,109,118,127,136,
%T A297268 145,154,164,173,182,191,200,209,218,227,236,246,255,264,273,282,291,
%U A297268 300,309,318,328,337,346,355,364,373,382,391,400,410,419,428,437,446
%N A297268 Numbers whose base-9 digits have equal down-variation and up-variation; see Comments.
%C A297268 Suppose that n has base-b digits b(m), b(m-1), ..., b(0).  The base-b down-variation of n is the sum DV(n,b) of all d(i)-d(i-1) for which d(i) > d(i-1); the base-b up-variation of n is the sum UV(n,b) of all d(k-1)-d(k) for which d(k) < d(k-1).  The total base-b variation of n is the sum TV(n,b) = DV(n,b) + UV(n,b).  See the guide at A297330.
%C A297268 Differs from A029955 first at 739=1011_9 which is not a palindrome in base 9 but has DV(739,9)=UV(793,9) =1. - _R. J. Mathar_, Jan 23 2018
%H A297268 Clark Kimberling, <a href="/A297268/b297268.txt">Table of n, a(n) for n = 1..10000</a>
%e A297268 446 in base-9:  5,4,5, having DV = 1, UV = 1, so that 446 is in the sequence.
%t A297268 g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
%t A297268 x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
%t A297268 b = 9; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
%t A297268 w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
%t A297268 Take[Flatten[Position[w, -1]], 120]   (* A297267 *)
%t A297268 Take[Flatten[Position[w, 0]], 120]    (* A297268 *)
%t A297268 Take[Flatten[Position[w, 1]], 120]    (* A297269 *)
%Y A297268 Cf. A297330, A297267, A297269.
%K A297268 nonn,base,easy
%O A297268 1,2
%A A297268 _Clark Kimberling_, Jan 15 2018