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A297030 Number of pieces in the list d(m),d(m-1),...,d(0) of base-2 digits of n; see Comments 17
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

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

LINKS

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

EXAMPLE

Base-2 digits for 100:  1, 1, 0, 0, 1, 0, 0, so that a(100) = 6.

MATHEMATICA

a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];

b = 2; Table[a[n, b], {n, 1, 120}]

CROSSREFS

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

KEYWORD

nonn,easy,base

AUTHOR

Clark Kimberling, Jan 13 2018

STATUS

approved

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Last modified January 17 23:00 EST 2021. Contains 340247 sequences. (Running on oeis4.)