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A329745
Number of compositions of n with runs-resistance 2.
20
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
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.
Sequence in context: A293534 A066653 A081945 * A255353 A248652 A158027
KEYWORD
nonn
AUTHOR
Gus Wiseman, Nov 21 2019
EXTENSIONS
Terms a(21) and beyond from Andrew Howroyd, Dec 30 2020
STATUS
approved