A297030 Number of pieces in the list d(m), d(m-1), ..., d(0) of base-2 digits of n; see Comments.

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

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.

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).

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

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