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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.)