A335514 Number of (1,2,3)-matching compositions of n.

0, 0, 0, 0, 0, 0, 1, 4, 14, 42, 114, 292, 714, 1686, 3871, 8696, 19178, 41667, 89386, 189739, 399144, 833290, 1728374, 3565148, 7319212, 14965880, 30496302, 61961380, 125577752, 253971555, 512716564, 1033496947, 2080572090, 4183940550, 8406047907, 16875834728

Offset: 0

Author

Gus Wiseman, Jun 22 2020

Keywords

• nonn

Examples

			The a(6) = 1 through a(8) = 14 compositions:
  (1,2,3)  (1,2,4)    (1,2,5)
           (1,1,2,3)  (1,3,4)
           (1,2,1,3)  (1,1,2,4)
           (1,2,3,1)  (1,2,1,4)
                      (1,2,2,3)
                      (1,2,3,2)
                      (1,2,4,1)
                      (2,1,2,3)
                      (1,1,1,2,3)
                      (1,1,2,1,3)
                      (1,1,2,3,1)
                      (1,2,1,1,3)
                      (1,2,1,3,1)
                      (1,2,3,1,1)
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..500
• Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Crossrefs

The version for permutations is A056986.

The avoiding version is A102726.

These compositions are ranked by A335479.

The version for patterns is A335515.

The version for prime indices is A335520.

Permutations are counted by A000142 and ranked by A333218.

Patterns are counted by A000670 and ranked by A333217.

Patterns matched by compositions are counted by A335456.

Cf. A011782, A032020, A106356, A226316, A269134, A333755, A335465, A335521.

Programs

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x
				

Formula

a(n > 0) = 2^(n - 1) - A102726(n).

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 31 2020