A297245 - OEIS

%I #4 Jan 17 2018 17:52:36

%S 0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,1,2,3,4,5,6,7,8,9,10,11,12,13,2,1,0,

%T 1,2,3,4,5,6,7,8,9,10,11,12,3,2,1,0,1,2,3,4,5,6,7,8,9,10,11,4,3,2,1,0,

%U 1,2,3,4,5,6,7,8,9,10,5,4,3,2,1,0,1,2,3

%N Total variation of base-15 digits of n; see Comments.

%C 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).  See A297330 for a guide to related sequences and partitions of the natural numbers:

%H Clark Kimberling, <a href="/A297245/b297245.txt">Table of n, a(n) for n = 1..10000</a>

%e 2^20 in base 15:  1, 5, 10, 10, 5, 1; here, DV = 9 and UV = 9, so that a(2^20) = 18.

%t b = 15; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &,      Partition[IntegerDigits[n, b], 2, 1]], {n, z}] (* cf. Michael De Vlieger, e.g. A037834 *)

%Y Cf. A297243, A297244, A297330.

%K nonn,base,easy

%O 1,18

%A _Clark Kimberling_, Jan 17 2018