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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.

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,changed

AUTHOR

Clark Kimberling, Jan 13 2018

STATUS

approved

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Last modified January 24 12:26 EST 2022. Contains 350537 sequences. (Running on oeis4.)