A329743 Number of compositions of n with runs-resistance n - 3.

0, 0, 0, 1, 2, 6, 9, 16, 8

Offset

0,5

Comments

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

Links

Table of n, a(n) for n=0..8.

Example

The a(3) = 1 through a(8) = 8 compositions:
  (3)  (22)    (14)   (114)    (1123)    (12113)
       (1111)  (23)   (411)    (1132)    (12212)
               (32)   (1113)   (1141)    (13112)
               (41)   (1221)   (1411)    (21131)
               (131)  (2112)   (2122)    (21221)
               (212)  (3111)   (2212)    (31121)
                      (11112)  (2311)    (121112)
                      (11211)  (3211)    (211121)
                      (21111)  (11131)
                               (11212)
                               (11221)
                               (12211)
                               (13111)
                               (21211)
                               (111121)
                               (121111)
For example, repeatedly taking run-lengths starting with (1,2,1,1,3) gives (1,2,1,1,3) -> (1,1,2,1) -> (2,1,1) -> (1,2) -> (1,1) -> (2), which is 5 steps, and 5 = 8 - 3, so (1,2,1,1,3) is counted under a(8).

Mathematica

runsres[q_]:=If[Length[q]==1, 0, Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==n-3&]], {n, 10}]

Crossrefs

Column k = n - 3 of A329744.

Column k = 3 of A329750.

Compositions with runs-resistance 2 are A329745.

Cf. A000740, A008965, A098504, A242882, A318928, A329746, A329747, A329767.

Sequence in context: A345051 A265202 A355738 * A320496 A172433 A049622

Adjacent sequences: A329740 A329741 A329742 * A329744 A329745 A329746

Keyword

nonn,fini,full

Author

Gus Wiseman, Nov 21 2019

Status

approved