cp's OEIS Frontend

A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.

%I A321773 #25 Aug 18 2025 00:38:03
%S A321773 1,3,6,4,9,9,10,12,15,13,18,18,19,21,24,22,27,27,28,30,33,31,36,36,37,
%T A321773 39,42,40,45,45,46,48,51,49,54,54,55,57,60,58,63,63,64,66,69,67,72,72,
%U A321773 73,75,78,76,81,81,82,84,87,85,90,90,91,93,96,94,99,99
%N A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts.
%H A321773 Alois P. Heinz, <a href="/A321773/b321773.txt">Table of n, a(n) for n = 3..1000</a>
%F A321773 Conjectures from _Colin Barker_, Dec 11 2018: (Start)
%F A321773 G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).
%F A321773 a(n) = a(n-2) + a(n-3) - a(n-5) for n>7. (End)
%F A321773 Conjecture: a(n) = (3*n-k)/2 where k value has a cycle of 6 starting from n=3 of (7,6,3,10,3,6). - _Bill McEachen_, Aug 12 2025
%e A321773 From _Gus Wiseman_, Nov 11 2020: (Start)
%e A321773 Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
%e A321773   111   112   113   114   115   116   117   118   119
%e A321773         121   122   141   133   161   144   181   155
%e A321773         211   131   222   151   224   171   226   191
%e A321773               212   411   223   233   225   244   227
%e A321773               221         232   242   252   262   272
%e A321773               311         313   323   333   334   335
%e A321773                           322   332   414   343   344
%e A321773                           331   422   441   424   353
%e A321773                           511   611   522   433   434
%e A321773                                       711   442   443
%e A321773                                             622   515
%e A321773                                             811   533
%e A321773                                                   551
%e A321773                                                   722
%e A321773                                                   911
%e A321773 (End)
%t A321773 Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* _Gus Wiseman_, Nov 11 2020 *)
%Y A321773 Column k=3 of A242887.
%Y A321773 A235451 counts 3-part compositions with distinct run-lengths
%Y A321773 A001399(n-6) counts 3-part compositions in the complement.
%Y A321773 A014311 intersected with A335488 ranks these compositions.
%Y A321773 A140106 is the unordered case, with Heinz numbers A285508.
%Y A321773 A261982 counts non-strict compositions of any length.
%Y A321773 A001523 counts unimodal compositions, with complement A115981.
%Y A321773 A007318 and A097805 count compositions by length.
%Y A321773 A032020 counts strict compositions.
%Y A321773 A047967 counts non-strict partitions, with Heinz numbers A013929.
%Y A321773 A242771 counts triples that are not strictly increasing.
%Y A321773 Cf. A000212, A000217, A001840, A128422, A156040, A332834, A337461, A337484, A337603, A337604.
%K A321773 nonn,changed
%O A321773 3,2
%A A321773 _Alois P. Heinz_, Nov 18 2018