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%I #11 Jan 18 2022 02:39:12
%S 0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,
%U 2,2,2,2,1,2,2,2,2,2,2,2,2,2,1,1,1,1
%N Number of pieces in the list d(m), d(m-1), ..., d(0) of base-8 digits of n; see Comments.
%C 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. See A297030 for a guide to related sequences.
%H Clark Kimberling, <a href="/A297036/b297036.txt">Table of n, a(n) for n = 1..10000</a>
%e Base-8 digits for 12345: 3, 0, 0, 7, 1, so that a(12345) = 4.
%t a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
%t b = 8; Table[a[n, b], {n, 1, 120}]
%Y Cf. A297030 (pieces), A296712 (rises and falls), A296882 (pits and peaks).
%K nonn,easy,base
%O 1,64
%A _Clark Kimberling_, Jan 13 2018
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