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 A297041 Number of pieces in the list d(m),d(m-1),...,d(0) of base-13 digits of n; see Comments 2

%I

%S 0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%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-13 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="/A297041/b297041.txt">Table of n, a(n) for n = 1..10000</a>

%e Base-13 digits for 1234567: 3, 4, 2, 12, 1, 9, so that a(124567) = 5.

%t a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];

%t b = 13; 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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Last modified May 13 11:42 EDT 2021. Contains 343839 sequences. (Running on oeis4.)