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

%I A329745 #9 Dec 30 2020 19:44:02
%S A329745 0,0,2,3,6,15,22,41,72,129,213,395,660,1173,2031,3582,6188,10927,
%T A329745 18977,33333,58153,101954,178044,312080,545475,955317,1670990,2925711,
%U A329745 5118558,8960938,15680072,27447344,48033498,84076139,147142492,257546234,450748482,788937188
%N A329745 Number of compositions of n with runs-resistance 2.
%C A329745 A composition of n is a finite sequence of positive integers with sum n.
%C A329745 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.
%C A329745 These are non-constant compositions with equal run-lengths (A329738).
%H A329745 Andrew Howroyd, <a href="/A329745/b329745.txt">Table of n, a(n) for n = 1..1000</a>
%F A329745 a(n) = A329738(n) - A000005(n).
%F A329745 a(n) = Sum_{d|n} (A003242(d) - 1). - _Andrew Howroyd_, Dec 30 2020
%e A329745 The a(3) = 2 through a(6) = 15 compositions:
%e A329745   (1,2)  (1,3)    (1,4)    (1,5)
%e A329745   (2,1)  (3,1)    (2,3)    (2,4)
%e A329745          (1,2,1)  (3,2)    (4,2)
%e A329745                   (4,1)    (5,1)
%e A329745                   (1,3,1)  (1,2,3)
%e A329745                   (2,1,2)  (1,3,2)
%e A329745                            (1,4,1)
%e A329745                            (2,1,3)
%e A329745                            (2,3,1)
%e A329745                            (3,1,2)
%e A329745                            (3,2,1)
%e A329745                            (1,1,2,2)
%e A329745                            (1,2,1,2)
%e A329745                            (2,1,2,1)
%e A329745                            (2,2,1,1)
%t A329745 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329745 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==2&]],{n,10}]
%o A329745 (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
%Y A329745 Column k = 2 of A329744.
%Y A329745 Column k = n - 2 of A329750.
%Y A329745 Cf. A000740, A003242, A008965, A098504, A242882, A318928, A329743, A329746, A329747, A329767.
%K A329745 nonn
%O A329745 1,3
%A A329745 _Gus Wiseman_, Nov 21 2019
%E A329745 Terms a(21) and beyond from _Andrew Howroyd_, Dec 30 2020