%I #4 Apr 22 2021 21:53:37
%S 0,0,0,0,0,1,0,2,2,3,4,7,5,9,11,12,14,18,18,23,23,27,32,37,35,40,46,
%T 49,53,59,60,68,71,76,83,88,90,99,104,111,115,123,128,137,142,149,160,
%U 167,168,177,187,194,205,215,219,229,235,247,260,269,271,285,294,304,313
%N Number of partitions of n into 3 distinct parts [r,s,t] such that r < s < t where at least 1 part does not divide any larger part.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((nj)/2)} sign(c(i/j) + c((nij)/i) + c((nij)/j)) * (1  [j = i]) * (1  [nj = 2*i]), where c(n) = ceiling(n)  floor(n) and [ ] is the Iverson bracket.
%e a(9) = 2; [1,3,5], [2,3,4] (Not counted: [1,1,7], [1,2,6], [1,4,4], [2,2,5], [3,3,3]).
%t Table[Sum[Sum[(1  KroneckerDelta[i, j]) (1  KroneckerDelta[2 i, n  j]) Sign[(Ceiling[i/j]  Floor[i/j]) + (Ceiling[(n  i  j)/j]  Floor[(n  i  j)/j]) + (Ceiling[(n  i  j)/i]  Floor[(n  i  j)/i])], {i, j, Floor[(n  j)/2]}], {j, Floor[n/3]}], {n, 100}]
%Y Cf. A343246, A343396.
%K nonn
%O 1,8
%A _Wesley Ivan Hurt_, Apr 13 2021
