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.
%I A353865 #12 Feb 21 2025 05:32:18
%S A353865 1,1,1,2,1,2,2,2,2,3,2,2,2,3,2,5,2,3,4,3,2,4,3,3,4,4,3,4,3,4,5,5,4,6,
%T A353865 4,6,5,4,5,6,5,6,7,6,5,9,6,6,7,6,8,9,6,6,8,9,7,9,9,7,10,9,8,13,7,10,
%U A353865 11,8,9,10,11,12,9,11,9,15,12,12,19,13,16,16
%N A353865 Number of complete rucksack partitions of n. Partitions whose weak run-sums are distinct and cover an initial interval of nonnegative integers.
%C A353865 Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). A weak run-sum is the sum of any consecutive constant subsequence.
%C A353865 Do all positive integers appear only finitely many times in this sequence?
%e A353865 The a(n) compositions for n = 1, 3, 9, 15, 18:
%e A353865 (1) (21) (4311) (54321) (543321)
%e A353865 (111) (51111) (532221) (654111)
%e A353865 (111111111) (651111) (7611111)
%e A353865 (81111111) (111111111111111111)
%e A353865 (111111111111111)
%e A353865 For example, the weak runs of y = {7,5,4,4,3,3,3,1,1} are {}, {1}, {1,1}, {3}, {4}, {5}, {3,3}, {7}, {4,4}, {3,3,3}, with sums 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, which are all distinct and cover an initial interval, so y is counted under a(31).
%t A353865 norqQ[m_]:=Sort[m]==Range[0,Max[m]];
%t A353865 msubs[s_]:=Join@@@Tuples[Table[Take[t,i],{t,Split[s]},{i,0,Length[t]}]];
%t A353865 Table[Length[Select[IntegerPartitions[n],norqQ[Total/@Select[msubs[#],SameQ@@#&]]&]],{n,0,15}]
%o A353865 (PARI) a(n) = my(c=0, s, v); if(n, forpart(p=n, if(p[1]==1, v=List([s=1]); for(i=2, #p, if(p[i]==p[i-1], listput(v, s+=p[i]), listput(v, s=p[i]))); s=#v; listsort(v, 1); if(s==#v&&s==v[s], c++))); c, 1); \\ _Jinyuan Wang_, Feb 21 2025
%Y A353865 Perfect partitions are counted by A002033, ranked by A325780.
%Y A353865 Knapsack partitions are counted by A108917, ranked by A299702.
%Y A353865 This is the complete case of A353864, ranked by A353866.
%Y A353865 These partitions are ranked by A353867.
%Y A353865 A000041 counts partitions, strict A000009.
%Y A353865 A275870 counts collapsible partitions, ranked by A300273.
%Y A353865 A304442 counts partitions with all equal run-sums, ranked by A353833.
%Y A353865 A353832 represents the operation of taking run-sums of a partition.
%Y A353865 A353836 counts partitions by number of distinct run-sums.
%Y A353865 A353837 counts partitions with distinct run-sums, ranked by A353838.
%Y A353865 A353840-A353846 pertain to partition run-sum trajectory.
%Y A353865 A353850 counts compositions with all distinct run-sums, ranked by A353852.
%Y A353865 A353863 counts partitions whose weak run-sums cover an initial interval.
%Y A353865 Cf. A008284, A018818, A225485, A325239, A325862, A353834, A353839.
%K A353865 nonn
%O A353865 0,4
%A A353865 _Gus Wiseman_, Jun 04 2022
%E A353865 More terms from _Jinyuan Wang_, Feb 21 2025