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A297030
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Number of pieces in the list d(m),d(m-1),...,d(0) of base-2 digits of n; see Comments
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17
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0, 1, 1, 2, 2, 2, 1, 2, 3, 3, 3, 3, 3, 2, 1, 2, 3, 4, 4, 4, 4, 4, 3, 3, 4, 4, 4, 3, 3, 2, 1, 2, 3, 4, 4, 5, 5, 5, 4, 4, 5, 5, 5, 5, 5, 4, 3, 3, 4, 5, 5, 5, 5, 5, 4, 3, 4, 4, 4, 3, 3, 2, 1, 2, 3, 4, 4, 5, 5, 5, 4, 5, 6, 6, 6, 6, 6, 5, 4, 4, 5, 6, 6, 6, 6, 6
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OFFSET
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1,4
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COMMENTS
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The definition of "piece" starts with the base-b digits d(m), d(m-1), ..., d(0) of n. First, an *ascent* is a list (d(i),d(i-1),...,d(i-h)) such that d(i)<d(i-1)<...<d(i-h), where d(i+1)>=d(i) if i<m, and d(i-h-1)>=d(i-h) if i>h. A *descent* is a list (d(i),d(i-1),...,d(i-h)) such that d(i)>d(i-1)>...>d(i-h), where d(i+1)<=d(i) if i<m, and d(i-h-1)<=d(i-h) if i>h. A *flat* is a list (d(i),d(i-1),...,d(i-h)), where h>0, such that d(i)=d(i-1)=...=d(i-h), where d(i+1)!=d(i) if i<m, and d(i-h-1)!=d(i-h) if i>h. A *piece* is an ascent, a descent, or a flat. Example: 235621103 has five pieces: (2,3,5,6), (6,2,1), (1,1), (1,0), and (0,3); that's 2 ascents, 2 descents, and 1 flat. For every b, the "piece sequence" includes every positive integer infinitely many times.
Guide to related sequences:
***
Base # pieces for n>=1
2 A297030
3 A297031
4 A297032
5 A297033
6 A297034
7 A297035
8 A297036
9 A297037
10 A297038
11 A297039
12 A297040
13 A297041
14 A297042
15 A297043
16 A297044
20 A297045
60 A297046
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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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Base-2 digits for 100: 1, 1, 0, 0, 1, 0, 0, so that a(100) = 6.
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MATHEMATICA
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a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
b = 2; Table[a[n, b], {n, 1, 120}]
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CROSSREFS
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Cf. A297038, A296712 (rises and falls), A296882 (pits and peaks).
Sequence in context: A205011 A130790 A261904 * A266348 A179647 A029330
Adjacent sequences: A297027 A297028 A297029 * A297031 A297032 A297033
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KEYWORD
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nonn,easy,base
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AUTHOR
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Clark Kimberling, Jan 13 2018
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STATUS
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approved
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