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A297330
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Total variation of base-10 digits of n; see Comments.
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89
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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, 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, 3, 7, 6, 5, 4, 3, 2, 1, 0, 1, 2, 8, 7, 6, 5, 4, 3, 2
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OFFSET
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1,13
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COMMENTS
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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:
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Base b {DV(n,b)} {UV(n,b)} {TV(n,b)}
2 A033264 A037800 A037834
3 A037853 A037844 A037835
4 A037854 A037845 A037836
5 A037855 A037846 A037837
6 A037856 A037847 A037838
7 A037857 A037848 A037839
8 A037858 A037849 A037840
9 A037859 A037850 A037841
10 A037860 A037851 A297330
11 A297231 A297232 A297233
12 A297234 A297235 A297236
13 A297237 A297238 A297239
14 A297240 A297241 A297242
15 A297243 A297244 A297245
16 A297246 A297247 A297247
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:
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Base b Sequence u Sequence e Sequence d
2 A005843 A005408 (none)
3 A297249 A297250 A297251
4 A297252 A297253 A297254
5 A297255 A297256 A297257
6 A297258 A297259 A297260
7 A297261 A297262 A297263
8 A297264 A297265 A297266
9 A297267 A297268 A297269
10 A297270 A297271 A297272
11 A297273 A297274 A297275
12 A297276 A297277 A297278
13 A297279 A297280 A297281
14 A297282 A297283 A297284
15 A297285 A297286 A297287
16 A297288 A297289 A297290
Not a duplicate of A151950: e.g., a(100)=1 but A151950(100)=11. - Robert Israel, Feb 06 2018
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LINKS
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Clark Kimberling, Table of n, a(n) for n = 1..10000
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EXAMPLE
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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.
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MAPLE
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f:= proc(n) local L, i; L:= convert(n, base, 10);
add(abs(L[i+1]-L[i]), i=1..nops(L)-1) end proc:
map(f, [$1..100]); # Robert Israel, Feb 04 2018
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MATHEMATICA
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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 *)
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CROSSREFS
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Cf. A037851, A297330, A297271, A297272.
Sequence in context: A255594 A030108 A307651 * A037904 A070615 A040114
Adjacent sequences: A297327 A297328 A297329 * A297331 A297332 A297333
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KEYWORD
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nonn,base,easy
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AUTHOR
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Clark Kimberling, Jan 17 2018
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STATUS
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approved
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