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%I #8 Nov 04 2018 01:44:47
%S 1,2,3,4,5,6,7,9,18,27,36,45,54,63,65,73,81,89,97,105,113,121,130,138,
%T 146,154,162,170,178,186,195,203,211,219,227,235,243,251,260,268,276,
%U 284,292,300,308,316,325,333,341,349,357,365,373,381,390,398,406
%N Numbers whose base-8 digits have equal down-variation and up-variation; see Comments.
%C 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 a(n) = A029803(n+1) for 1 <= n < 72, but a(72) = 521 differs from A029803(73) = 585. - _Georg Fischer_, Oct 30 2018
%H Clark Kimberling, <a href="/A297265/b297265.txt">Table of n, a(n) for n = 1..10000</a>
%e 406 in base-8: 6,2,6, having DV = 4, UV = 4, so that 406 is in the sequence.
%t g[n_, b_] := Map[Total, GatherBy[Differences[IntegerDigits[n, b]], Sign]];
%t x[n_, b_] := Select[g[n, b], # < 0 &]; y[n_, b_] := Select[g[n, b], # > 0 &];
%t b = 8; z = 2000; p = Table[x[n, b], {n, 1, z}]; q = Table[y[n, b], {n, 1, z}];
%t w = Sign[Flatten[p /. {} -> {0}] + Flatten[q /. {} -> {0}]];
%t Take[Flatten[Position[w, -1]], 120] (* A297264 *)
%t Take[Flatten[Position[w, 0]], 120] (* A297265 *)
%t Take[Flatten[Position[w, 1]], 120] (* A297266 *)
%Y Cf. A297330, A297264, A297266.
%K nonn,base,easy
%O 1,2
%A _Clark Kimberling_, Jan 15 2018
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