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%I #17 Nov 12 2020 22:19:28
%S 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 39,42,40,45,45,46,48,51,49,54,54,55,57,60,58,63,63,64,66,69,67,72,72,
%U 73,75,78,76,81,81,82,84,87,85,90,90,91,93,96,94,99,99
%N Number of compositions of n into parts with distinct multiplicities and with exactly three parts.
%H Alois P. Heinz, <a href="/A321773/b321773.txt">Table of n, a(n) for n = 3..1000</a>
%F Conjectures from _Colin Barker_, Dec 11 2018: (Start)
%F G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).
%F a(n) = a(n-2) + a(n-3) - a(n-5) for n>7.
%F (End)
%e From _Gus Wiseman_, Nov 11 2020: (Start)
%e Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
%e 111 112 113 114 115 116 117 118 119
%e 121 122 141 133 161 144 181 155
%e 211 131 222 151 224 171 226 191
%e 212 411 223 233 225 244 227
%e 221 232 242 252 262 272
%e 311 313 323 333 334 335
%e 322 332 414 343 344
%e 331 422 441 424 353
%e 511 611 522 433 434
%e 711 442 443
%e 622 515
%e 811 533
%e 551
%e 722
%e 911
%e (End)
%t Table[Length[Join@@Permutations/@Select[IntegerPartitions[n,{3}],!UnsameQ@@#&]],{n,0,100}] (* _Gus Wiseman_, Nov 11 2020 *)
%Y Column k=3 of A242887.
%Y A235451 counts 3-part compositions with distinct run-lengths
%Y A001399(n-6) counts 3-part compositions in the complement.
%Y A014311 intersected with A335488 ranks these compositions.
%Y A140106 is the unordered case, with Heinz numbers A285508.
%Y A261982 counts non-strict compositions of any length.
%Y A001523 counts unimodal compositions, with complement A115981.
%Y A007318 and A097805 count compositions by length.
%Y A032020 counts strict compositions.
%Y A047967 counts non-strict partitions, with Heinz numbers A013929.
%Y A242771 counts triples that are not strictly increasing.
%Y Cf. A000212, A000217, A001840, A128422, A156040, A332834, A337461, A337484, A337603, A337604.
%K nonn
%O 3,2
%A _Alois P. Heinz_, Nov 18 2018
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