This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
[ #&cat RestrictedPartitions(n,{3..n}):n in [0..50]];
Table[Length[Flatten[Select[IntegerPartitions[n],Min[#]>2&]]],{n,0,50}] (* Harvey P. Dale, May 12 2020 *)
For n = 8 the seven partitions of 8 that do not contain 1 as a part are: . 8 . 4 + 4 . (5) + 3 . (6) + 2 . (3) + (3) + 2 . (4) + 2 + 2 . 2 + 2 + 2 + 2 Note that in every partition the parts that are not the smallest part are shown between parentheses. The total number of parts that are not the smallest part is 0+0+1+1+2+1+0 = 5, so a(8) = 5.
For n = 9 the diagram of
the partitions of 9 that
do not contain 1 as a part
is as shown below: Partitions
.
|_ _ _| | | | [3, 2, 2, 2]
|_ _ _ _ _| | | [5, 2, 2]
|_ _ _ _| | | [4, 3, 2]
|_ _ _ _ _ _ _| | [7, 2]
|_ _ _| | | [3, 3, 3]
|_ _ _ _ _ _| | [6, 3]
|_ _ _ _ _| | [5, 4]
|_ _ _ _ _ _ _ _ _| [9]
.
Note that the above diagram is also the "head" of the last section of the set of partitions of 9, where the "tail" is formed by A000041(9-1)= 22 1's.
The diagram of the
emergent parts is as
shown below: Emergent parts
.
|_ _ _| | | [3, 2, 2]
|_ _ _ _ _| | [5, 2]
|_ _ _ _| | [4, 3]
|_ _ _ _ _ _ _| [7]
|_ _ _| | [3, 3]
|_ _ _ _ _ _| [6]
|_ _ _ _ _| [5]
.
The sum of the largest emergent parts is 3 + 5 + 4 + 7 + 3 + 6 + 5 = 33, so a(9) = 33.
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