A297045 - OEIS

%I #7 Jan 14 2018 18:23:12

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

%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-20 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="/A297045/b297045.txt">Table of n, a(n) for n = 1..10000</a>

%F Base-20 digits for 123456789:  1, 18, 11, 12, 1, 19, 9, so that a(12456789) = 6.

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

%t b = 20; 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