A228371 First differences of A228370. Also A001511 and A006519 interleaved.

1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 5, 16, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 6, 32, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 5, 16, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 4, 8, 1, 1, 2, 2, 1, 1, 3, 4, 1, 1, 2, 2, 1, 1, 7, 64

Offset: 1

Author

Omar E. Pol, Aug 21 2013

Keywords

• nonn
• tabf

Comments

Number of toothpicks added at n-th stage to the toothpick structure (related to integer compositions) of A228370.

The equivalent sequence for integer partitions is A220517.

Examples

			Illustration of the structure after 32 stages. The diagram represents the 16 compositions of 5. The k-th horizontal line segment has length A001511(k) equals the largest part of the k-th region. The k-th vertical line segment has length A006519(k) equals the number of parts of the k-th region.
.      _ _ _ _ _
16     _        |
15     _|_      |
14     _  |     |
13     _|_|_    |
12     _    |   |
11     _|_  |   |
10     _  | |   |
9      _|_|_|_  |
8      _      | |
7      _|_    | |
6      _  |   | |
5      _|_|_  | |
4      _    | | |
3      _|_  | | |
2      _  | | | |
1       | | | | |
.
Written as an irregular triangle the sequence begins:
  1,1;
  2,2;
  1,1,3,4;
  1,1,2,2,1,1,4,8;
  1,1,2,2,1,1,3,4,1,1,2,2,1,1,5,16;
  1,1,2,2,1,1,3,4,1,1,2,2,1,1,4,8,1,1,2,2,1,1,3,4,1,1,2,2,1,1,6,32;
  ...
		

Links

• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Row lengths give 2*A011782. Right border gives A000079.

Cf. A001511, A006519, A139250, A139251, A206437, A220517, A228370.

Programs

Python

def A228371(n): return ((m:=(n>>1)+1)&-m).bit_length() if n&1 else (m:=n>>1)&-m # Chai Wah Wu, Jul 14 2022

Formula

a(2n-1) = A001511(n), n >= 1. a(2n) = A006519(n), n >= 1.