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A297237 Down-variation of the base-13 digits of n; see Comments.

%I A297237 #7 Jan 19 2018 16:22:38
%S A297237 0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,2,1,0,0,0,0,0,0,0,
%T A297237 0,0,0,0,3,2,1,0,0,0,0,0,0,0,0,0,0,4,3,2,1,0,0,0,0,0,0,0,0,0,5,4,3,2,
%U A297237 1,0,0,0,0,0,0,0,0,6,5,4,3,2,1,0,0,0
%N A297237 Down-variation of the base-13 digits of n; see Comments.
%C A297237 Suppose that a number 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). Every positive integer occurs infinitely many times. See A297330 for a guide to related sequences and partitions of the natural numbers.
%H A297237 Clark Kimberling, <a href="/A297237/b297237.txt">Table of n, a(n) for n = 1..10000</a>
%e A297237 26 in base 13: 2,0; here DV = 2, so that a(26) = 2.
%t A297237 g[n_, b_] := Differences[IntegerDigits[n, b]];
%t A297237 b = 13; z = 120; Table[-Total[Select[g[n, b], # < 0 &]], {n, 1, z}];  (* A297237 *)
%t A297237 Table[Total[Select[g[n, b], # > 0 &]], {n, 1, z}]; (* A297238 *)
%Y A297237 Cf. A297238, A297239, A297330.
%K A297237 nonn,base,easy
%O A297237 1,26
%A A297237 _Clark Kimberling_, Jan 17 2018