A329745 Number of compositions of n with runs-resistance 2.

0, 0, 2, 3, 6, 15, 22, 41, 72, 129, 213, 395, 660, 1173, 2031, 3582, 6188, 10927, 18977, 33333, 58153, 101954, 178044, 312080, 545475, 955317, 1670990, 2925711, 5118558, 8960938, 15680072, 27447344, 48033498, 84076139, 147142492, 257546234, 450748482, 788937188

Offset

1,3

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.

These are non-constant compositions with equal run-lengths (A329738).

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Formula

a(n) = A329738(n) - A000005(n).

a(n) = Sum_{d|n} (A003242(d) - 1). - Andrew Howroyd, Dec 30 2020

Example

The a(3) = 2 through a(6) = 15 compositions:
  (1,2)  (1,3)    (1,4)    (1,5)
  (2,1)  (3,1)    (2,3)    (2,4)
         (1,2,1)  (3,2)    (4,2)
                  (4,1)    (5,1)
                  (1,3,1)  (1,2,3)
                  (2,1,2)  (1,3,2)
                           (1,4,1)
                           (2,1,3)
                           (2,3,1)
                           (3,1,2)
                           (3,2,1)
                           (1,1,2,2)
                           (1,2,1,2)
                           (2,1,2,1)
                           (2,2,1,1)

Mathematica

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&, q, Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], runsres[#]==2&]], {n, 10}]

Prog

(PARI) seq(n)={my(b=Vec(1/(1 - sum(k=1, n, x^k/(1+x^k) + O(x*x^n)))-1)); vector(n, k, sumdiv(k, d, b[d]-1))} \\ Andrew Howroyd, Dec 30 2020

Crossrefs

Column k = 2 of A329744.

Column k = n - 2 of A329750.

Cf. A000740, A003242, A008965, A098504, A242882, A318928, A329743, A329746, A329747, A329767.

Sequence in context: A293534 A066653 A081945 * A255353 A248652 A158027

Adjacent sequences: A329742 A329743 A329744 * A329746 A329747 A329748

Keyword

nonn

Author

Gus Wiseman, Nov 21 2019

Extensions

Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020

Status

approved