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A297253 Numbers whose base-4 digits having equal up-variation and down-variation; see Comments.

%I A297253 #4 Jan 15 2018 15:31:48
%S A297253 1,2,3,5,10,15,17,21,25,29,34,38,42,46,51,55,59,63,65,69,73,77,81,85,
%T A297253 89,93,97,101,105,109,113,117,121,125,130,134,138,142,146,150,154,158,
%U A297253 162,166,170,174,178,182,186,190,195,199,203,207,211,215,219,223
%N A297253 Numbers whose base-4 digits having equal up-variation and down-variation; see Comments.
%C A297253 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.
%H A297253 Clark Kimberling, <a href="/A297253/b297253.txt">Table of n, a(n) for n = 1..10000</a>
%e A297253 223 in base-4:  3,2,3,3, having DV = 1, UV = 1, so that 223 is in the sequence.
%t A297253 g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
%t A297253 x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
%t A297253 b = 4; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
%t A297253 w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
%t A297253 Take[Flatten[Position[w, -1]], 120]   (* A297252 *)
%t A297253 Take[Flatten[Position[w, 0]], 120]    (* A297253 *)
%t A297253 Take[Flatten[Position[w, 1]], 120]    (* A297254 *)
%Y A297253 Cf. A297330, A297252, A297254.
%K A297253 nonn,base,easy
%O A297253 1,2
%A A297253 _Clark Kimberling_, Jan 15 2018