cp's OEIS Frontend

A297030 Number of pieces in the list d(m), d(m-1), ..., d(0) of base-2 digits of n; see Comments.

%I A297030 #15 Jan 18 2022 05:37:05
%S A297030 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,
%T A297030 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,
%U A297030 5,5,4,5,6,6,6,6,6,5,4,4,5,6,6,6,6,6
%N A297030 Number of pieces in the list d(m), d(m-1), ..., d(0) of base-2 digits of n; see Comments.
%C A297030 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.
%H A297030 Clark Kimberling, <a href="/A297030/b297030.txt">Table of n, a(n) for n = 1..10000</a>
%e A297030 Base-2 digits for 100:  1, 1, 0, 0, 1, 0, 0, so that a(100) = 6.
%t A297030 a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
%t A297030 b = 2; Table[a[n, b], {n, 1, 120}]
%Y A297030 Cf. A297038, A296712 (rises and falls), A296882 (pits and peaks).
%Y A297030 Guide to related sequences:
%Y A297030 Base   # pieces for n >= 1
%Y A297030    2        A297030
%Y A297030    3        A297031
%Y A297030    4        A297032
%Y A297030    5        A297033
%Y A297030    6        A297034
%Y A297030    7        A297035
%Y A297030    8        A297036
%Y A297030    9        A297037
%Y A297030   10        A297038
%Y A297030   11        A297039
%Y A297030   12        A297040
%Y A297030   13        A297041
%Y A297030   14        A297042
%Y A297030   15        A297043
%Y A297030   16        A297044
%Y A297030   20        A297045
%Y A297030   60        A297046
%K A297030 nonn,easy,base
%O A297030 1,4
%A A297030 _Clark Kimberling_, Jan 13 2018