A297330 Total variation of base-10 digits of n; see Comments.

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

Offset: 1

Author

Clark Kimberling, Jan 17 2018

Keywords

• nonn
• base
• easy

Comments

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:

***

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

Examples

			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.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A037851, A297330, A297271, A297272.

Programs

Maple

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
# alternative
A297330 := proc(n)
    A037860(n)+A037851(n) ;
end proc: # R. J. Mathar, Sep 27 2021

Mathematica

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 *)

Python

def A297330(n):
    s = str(n)
    return sum(abs(int(s[i])-int(s[i+1])) for i in range(len(s)-1)) # Chai Wah Wu, May 31 2022