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%I #7 Jan 14 2018 18:22:47
%S 0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%T 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
%U 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1
%N Number of pieces in the list d(m), d(m-1), ..., d(0) of base-14 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="/A297042/b297042.txt">Table of n, a(n) for n = 1..10000</a>
%e Base-14 digits for 1234567: 2, 4, 1, 12, 11, 5, so that a(124567) = 4.
%t a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
%t b = 14; Table[a[n, b], {n, 1, 1000}]
%Y Cf. A297030 (pieces), A296712 (rises and falls), A296882 (pits and peaks).
%K nonn,easy,base
%O 1
%A _Clark Kimberling_, Jan 13 2018
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