|
|
A228366
|
|
Toothpick sequence from a diagram of compositions of the positive integers (see Comments lines for definition).
|
|
10
|
|
|
0, 2, 6, 8, 15, 17, 21, 23, 35, 37, 41, 43, 50, 52, 56, 58, 79, 81, 85, 87, 94, 96, 100, 102, 114, 116, 120, 122, 129, 131, 135, 137, 175, 177, 181, 183, 190, 192, 196, 198, 210, 212, 216, 218, 225, 227, 231, 233, 254, 256, 260, 262, 269, 271, 275
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
In order to construct this sequence we use the following rules:
We start in the first quadrant of the square grid with no toothpicks, so a(0) = 0.
At stage n we place the smallest possible number of toothpicks of length 1 connected by their endpoints in horizontal direction starting from the grid point (0, n) such that the x-coordinate of the exposed endpoint of the last toothpick is not equal to the x-coordinate of any outer corner of the structure. Then we place toothpicks of length 1 connected by their endpoints in vertical direction, starting from the exposed toothpick endpoint, downward up to touch the structure or up to touch the x-axis.
The sequence gives the number of toothpicks after n stages. A228367 (the first differences) gives the number of toothpicks added at the n-th stage.
Note that the number of toothpick of added at n-th stage in horizontal direction is also A001511(n) and the number of toothpicks added at n-th stage in vertical direction is also A006519(n). Also counting both the x-axis and the y-axis we have that A001511(n) is also the largest part of the n-th region of the diagram and A006519(n) is also the number of parts of the n-th region of the diagram.
After 2^k stages a new section of the structure is completed, so the structure can be interpreted as a diagram of the 2^(k-1) compositions of k in colexicographic order, if k >= 1 (see A228525). The infinite diagram can be interpreted as a table of compositions of the positive integers.
|
|
LINKS
|
|
|
FORMULA
|
|
|
EXAMPLE
|
Illustration of initial terms (n = 1..8):
. _ _ _ _
. _ _ |
. _ _ _|_ _|_ |
. _ _ | _ | _ | |
. _ _ _ _|_ _ _|_|_ _|_|_ _|_|_ |
. _ _ | _ | _ | _ | _ | |
. _ _ _|_ _|_ | _|_ | _|_ | _|_ | _|_ | |
._ _ | _ | _ | | _ | | _ | | _ | | _ | | |
. | | | | | | | | | | | | | | | | | | | | |
.
.2 6 8 15 17 21 23 35
.
After 16 stages there are 79 toothpicks in the structure which represents the compositions of 5 in colexicographic order as shown below:
-------------------------------
n Diagram Composition
-------------------------------
. _ _ _ _ _
16 _ | 5
15 _|_ | 1+4
14 _ | | 2+3
13 _|_|_ | 1+1+3
12 _ | | 3+2
11 _|_ | | 1+2+2
10 _ | | | 2+1+2
9 _|_|_|_ | 1+1+1+2
8 _ | | 4+1
7 _|_ | | 1+3+1
6 _ | | | 2+2+1
5 _|_|_ | | 1+1+2+1
4 _ | | | 3+1+1
3 _|_ | | | 1+2+1+1
2 _ | | | | 2+1+1+1
1 | | | | | 1+1+1+1+1
.
|
|
PROG
|
(Python)
def A228366(n): return sum(((m:=(i>>1)+1)&-m).bit_length() if i&1 else (m:=i>>1)&-m for i in range(1, 2*n+1)) # Chai Wah Wu, Jul 15 2022
|
|
CROSSREFS
|
Cf. A001511, A005187, A006519, A011782, A001792, A065120, A139250, A228367, A228370, A228371, A228525.
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|