OFFSET
0,2
COMMENTS
Conjecture: The rows eventually become periodic with period n if extended further. For example, row n = 8 begins:
22, 15, 19, 17, 20, 17, 20, 16,
22, 17, 20, 17, 21, 17, 20, 17,
22, 17, 20, 17, 21, 17, 20, 17, ...
EXAMPLE
Triangle begins:
1
2 1
3 2 2
5 3 4 3
7 5 6 6 6
11 7 9 8 9 7
15 11 13 13 14 13 14
22 15 19 17 20 17 20 16
30 22 26 26 27 26 28 26 27
42 30 37 34 39 33 40 34 39 34
56 42 50 49 52 50 54 51 54 53 53
77 56 68 64 71 63 73 63 71 65 70 62
101 77 91 89 95 90 97 93 97 97 98 94 99
135 101 122 115 127 115 130 114 131 119 130 117 132 116
176 135 159 156 165 157 170 161 167 168 166 165 172 164 166
Row n = 6 counts the following partitions:
(6) (51) (51) (51) (51) (51)
(51) (411) (42) (411) (42) (411)
(42) (321) (411) (33) (411) (321)
(411) (3111) (321) (321) (321) (3111)
(33) (2211) (3111) (3111) (3111) (2211)
(321) (21111) (222) (2211) (222) (21111)
(3111) (111111) (2211) (21111) (2211) (111111)
(222) (21111) (111111) (21111)
(2211) (111111) (111111)
(21111)
(111111)
MATHEMATICA
combu[n_, y_]:=With[{s=Table[{k, i}, {k, Union[y]}, {i, 0, Floor[n/k]}]}, Select[Tuples[s], Total[Times@@@#]==n&]];
Table[Length[Select[IntegerPartitions[n], combu[k, #]!={}&]], {n, 0, 12}, {k, 0, n-1}]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Sep 08 2023
STATUS
approved