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A321773 Number of compositions of n into parts with distinct multiplicities and with exactly three parts. 5
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, 39, 42, 40, 45, 45, 46, 48, 51, 49, 54, 54, 55, 57, 60, 58, 63, 63, 64, 66, 69, 67, 72, 72, 73, 75, 78, 76, 81, 81, 82, 84, 87, 85, 90, 90, 91, 93, 96, 94, 99, 99 (list; graph; refs; listen; history; text; internal format)
OFFSET
3,2
LINKS
FORMULA
Conjectures from Colin Barker, Dec 11 2018: (Start)
G.f.: x^3*(1 + 3*x + 5*x^2) / ((1 - x)^2*(1 + x)*(1 + x + x^2)).
a(n) = a(n-2) + a(n-3) - a(n-5) for n>7.
(End)
EXAMPLE
From Gus Wiseman, Nov 11 2020: (Start)
Also the number of 3-part non-strict compositions of n. For example, the a(3) = 1 through a(11) = 15 triples are:
111 112 113 114 115 116 117 118 119
121 122 141 133 161 144 181 155
211 131 222 151 224 171 226 191
212 411 223 233 225 244 227
221 232 242 252 262 272
311 313 323 333 334 335
322 332 414 343 344
331 422 441 424 353
511 611 522 433 434
711 442 443
622 515
811 533
551
722
911
(End)
MATHEMATICA
Table[Length[Join@@Permutations/@Select[IntegerPartitions[n, {3}], !UnsameQ@@#&]], {n, 0, 100}] (* Gus Wiseman, Nov 11 2020 *)
CROSSREFS
Column k=3 of A242887.
A235451 counts 3-part compositions with distinct run-lengths
A001399(n-6) counts 3-part compositions in the complement.
A014311 intersected with A335488 ranks these compositions.
A140106 is the unordered case, with Heinz numbers A285508.
A261982 counts non-strict compositions of any length.
A001523 counts unimodal compositions, with complement A115981.
A007318 and A097805 count compositions by length.
A032020 counts strict compositions.
A047967 counts non-strict partitions, with Heinz numbers A013929.
A242771 counts triples that are not strictly increasing.
Sequence in context: A021278 A336761 A143940 * A083349 A065230 A316478
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Nov 18 2018
STATUS
approved

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Last modified March 28 07:33 EDT 2024. Contains 371235 sequences. (Running on oeis4.)