A297038 Number of pieces in the list d(m), d(m-1), ..., d(0) of base-10 digits of n; see Comments.

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, 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, 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, 2, 2, 2, 2, 2

Offset: 1

Author

Clark Kimberling, Jan 13 2018

Keywords

• nonn
• easy
• base

Comments

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.

Examples

			Base-10 digits for 123321:  1,2,3,3,2,1, so that a(123321) = 3.
		

Links

• Clark Kimberling, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A297030 (pieces), A296712 (rises and falls), A296882 (pits and peaks).

Programs

Mathematica

a[n_, b_] := Length[Map[Length, Split[Sign[Differences[IntegerDigits[n, b]]]]]];
b = 10; Table[a[n, b], {n, 1, 1000}]