Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #23 Jan 07 2025 10:13:40
%S 1,2,3,2,4,5,3,6,3,7,4,8,9,5,4,10,4,11,6,12,5,13,7,14,5,15,8,6,5,16,
%T 17,9,18,7,6,19,10,20,6,21,11,8,6,22,7,23,12,24,9,25,13,7,26,8,27,14,
%U 10,7,28,7,29,15,30,11,9,8,31,16,32,33,17,12,8,34,10,35,18,9,8,36,13,8,37,19,38,11,39,20,14,9,40,10,41,21
%N Irregular triangle read by rows in which row n lists the largest parts of the partitions of n into consecutive parts (with the partitions ordered by increasing number of parts).
%C Row n gives the first A001227(n) terms of the n-th row of A379631.
%C Conjecture: the right border gives A212652.
%e Triangle begins:
%e 1;
%e 2;
%e 3, 2;
%e 4;
%e 5, 3;
%e 6, 3;
%e 7, 4;
%e 8;
%e 9, 5, 4;
%e 10, 4;
%e 11, 6;
%e 12, 5;
%e 13, 7;
%e 14, 5;
%e 15, 8, 6, 5;
%e 16;
%e 17, 9;
%e 18, 7, 6;
%e 19, 10;
%e 20, 6;
%e 21, 11, 8, 6;
%e 22, 7;
%e 23, 12;
%e 24, 9;
%e 25, 13, 7;
%e 26, 8;
%e 27, 14, 10, 7;
%e 28, 7;
%e ...
%e Illustration of initial terms:
%e _
%e _|1|
%e _|2 _|
%e _|3 |2|
%e _|4 _| |
%e _|5 |3 _|
%e _|6 _| |3|
%e _|7 |4 | |
%e _|8 _| _| |
%e _|9 |5 |4 _|
%e _|10 _| | |4|
%e _|11 |6 _| | |
%e _|12 _| |5 | |
%e _|13 |7 | _| |
%e _|14 _| _| |5 _|
%e _|15 |8 |6 | |5|
%e _|16 _| | | | |
%e _|17 |9 _| _| | |
%e _|18 _| |7 |6 | |
%e _|19 |10 | | _| |
%e _|20 _| _| | |6 _|
%e _|21 |11 |8 _| | |6|
%e _|22 _| | |7 | | |
%e _|23 |12 _| | | | |
%e _|24 _| |9 | _| | |
%e _|25 |13 | _| |7 | |
%e _|26 _| _| |8 | _| |
%e _|27 |14 |10 | | |7 _|
%e |28 | | | | | |7|
%e ...
%e The diagram is also the left part of the diagram of A379631.
%e The geometrical structure is the same as the diagram of A237591.
%Y Positive terms of A286013.
%Y Column 1 gives A000027.
%Y Row lengths give A001227.
%Y Row sums give A286015.
%Y Subsequence of A286000 and of A299765 and of A379631.
%Y For the smallest parts see A379632.
%Y Cf. A196020, A212652, A235791, A236104, A237048, A237591, A237593, A245092, A262626, A379630, A379634.
%K nonn,tabf,new
%O 1,2
%A _Omar E. Pol_, Dec 31 2024