login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A335513 Numbers k such that the k-th composition in standard order (A066099) avoids the pattern (1,1,1). 4
0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 32, 33, 34, 35, 36, 37, 38, 40, 41, 43, 44, 45, 46, 48, 49, 50, 52, 53, 54, 56, 58, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83, 84, 88, 89 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
These are compositions with no part appearing more than twice.
A composition of n is a finite sequence of positive integers summing to n. The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.
We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).
LINKS
EXAMPLE
The sequence of terms together with the corresponding compositions begins:
0: () 17: (4,1) 37: (3,2,1)
1: (1) 18: (3,2) 38: (3,1,2)
2: (2) 19: (3,1,1) 40: (2,4)
3: (1,1) 20: (2,3) 41: (2,3,1)
4: (3) 21: (2,2,1) 43: (2,2,1,1)
5: (2,1) 22: (2,1,2) 44: (2,1,3)
6: (1,2) 24: (1,4) 45: (2,1,2,1)
8: (4) 25: (1,3,1) 46: (2,1,1,2)
9: (3,1) 26: (1,2,2) 48: (1,5)
10: (2,2) 28: (1,1,3) 49: (1,4,1)
11: (2,1,1) 32: (6) 50: (1,3,2)
12: (1,3) 33: (5,1) 52: (1,2,3)
13: (1,2,1) 34: (4,2) 53: (1,2,2,1)
14: (1,1,2) 35: (4,1,1) 54: (1,2,1,2)
16: (5) 36: (3,3) 56: (1,1,4)
MATHEMATICA
stc[n_]:=Reverse[Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n, 2]], 1], 0]]];
Select[Range[0, 100], !MatchQ[stc[#], {___, x_, ___, x_, ___, x_, ___}]&]
CROSSREFS
These compositions are counted by A232432 (by sum).
The (1,1)-avoiding version is A233564.
The complement A335512 is the matching version.
Constant patterns are counted by A000005 and ranked by A272919.
Permutations are counted by A000142 and ranked by A333218.
Patterns are counted by A000670 and ranked by A333217.
Patterns avoiding (1,1,1) are counted by A080599 (by length).
Non-unimodal compositions are counted by A115981 and ranked by A335373.
Combinatory separations are counted by A269134.
Patterns matched by standard compositions are counted by A335454.
Minimal patterns avoided by a standard composition are counted by A335465.
Permutations of prime indices avoiding (1,1,1) are counted by A335511.
Sequence in context: A037474 A292638 A000378 * A367906 A022551 A172251
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 18 2020
STATUS
approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 4 16:59 EST 2024. Contains 370532 sequences. (Running on oeis4.)