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%I #26 May 31 2022 11:02:05
%S 0,0,0,0,0,0,0,0,0,1,0,1,2,3,4,5,6,7,8,2,1,0,1,2,3,4,5,6,7,3,2,1,0,1,
%T 2,3,4,5,6,4,3,2,1,0,1,2,3,4,5,5,4,3,2,1,0,1,2,3,4,6,5,4,3,2,1,0,1,2,
%U 3,7,6,5,4,3,2,1,0,1,2,8,7,6,5,4,3,2
%N Total variation of base-10 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). Guide to related sequences and partitions of the natural numbers:
%C ***
%C Base b {DV(n,b)} {UV(n,b)} {TV(n,b)}
%C 2 A033264 A037800 A037834
%C 3 A037853 A037844 A037835
%C 4 A037854 A037845 A037836
%C 5 A037855 A037846 A037837
%C 6 A037856 A037847 A037838
%C 7 A037857 A037848 A037839
%C 8 A037858 A037849 A037840
%C 9 A037859 A037850 A037841
%C 10 A037860 A037851 A297330
%C 11 A297231 A297232 A297233
%C 12 A297234 A297235 A297236
%C 13 A297237 A297238 A297239
%C 14 A297240 A297241 A297242
%C 15 A297243 A297244 A297245
%C 16 A297246 A297247 A297247
%C For each b, let u = {n : UV(n,b) < DV(n,b)}, e = {n : UV(n,b) = DV(n,b)}, and d = {n : UV(n,b) > DV(n,b)}. The sets u,e,d partition the natural numbers. A guide to the matching sequences for u, e, d follows:
%C ***
%C Base b Sequence u Sequence e Sequence d
%C 2 A005843 A005408 (none)
%C 3 A297249 A297250 A297251
%C 4 A297252 A297253 A297254
%C 5 A297255 A297256 A297257
%C 6 A297258 A297259 A297260
%C 7 A297261 A297262 A297263
%C 8 A297264 A297265 A297266
%C 9 A297267 A297268 A297269
%C 10 A297270 A297271 A297272
%C 11 A297273 A297274 A297275
%C 12 A297276 A297277 A297278
%C 13 A297279 A297280 A297281
%C 14 A297282 A297283 A297284
%C 15 A297285 A297286 A297287
%C 16 A297288 A297289 A297290
%C Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - _Robert Israel_, Feb 06 2018
%H Clark Kimberling, <a href="/A297330/b297330.txt">Table of n, a(n) for n = 1..10000</a>
%e 13684632 has DV = 8-4 + 6-3 + 3-2 = 8 and has UV = 3-1 + 6-3 + 8-6 + 6-4 = 9, so that a(13684632) = DV + UV = 17.
%p f:= proc(n) local L,i; L:= convert(n,base,10);
%p add(abs(L[i+1]-L[i]),i=1..nops(L)-1) end proc:
%p map(f, [$1..100]); # _Robert Israel_, Feb 04 2018
%p # alternative
%p A297330 := proc(n)
%p A037860(n)+A037851(n) ;
%p end proc: # _R. J. Mathar_, Sep 27 2021
%t b = 10; z = 120; t = Table[Total@Flatten@Map[Abs@Differences@# &, Partition[ IntegerDigits[n, b], 2, 1]], {n, z}] (* after _Michael De Vlieger_, e.g. A037834 *)
%o (Python)
%o def A297330(n):
%o s = str(n)
%o return sum(abs(int(s[i])-int(s[i+1])) for i in range(len(s)-1)) # _Chai Wah Wu_, May 31 2022
%Y Cf. A037851, A297330, A297271, A297272.
%K nonn,base,easy
%O 1,13
%A _Clark Kimberling_, Jan 17 2018
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