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 #38 Jun 20 2022 10:13:38
%S 1,2,1,2,3,4,2,3,5,1,2,3,6,3,4,7,8,2,3,4,4,5,9,1,2,3,4,10,5,6,11,3,4,
%T 5,12,6,7,13,2,3,4,5,14,1,2,3,4,5,4,5,6,7,8,15,16,8,9,17,3,4,5,6,5,6,
%U 7,18,9,10,19,2,3,4,5,6,20,1,2,3,4,5,6,6,7,8,10,11,21,4,5,6,7,22,11,12,23,7,8,9,24
%N Irregular triangle read by rows, T(n,k), n >= 1, k >= 1, in which row n lists in reverse order the partitions of n into consecutive parts.
%H Paolo Xausa, <a href="/A328365/b328365.txt">Table of n, a(n) for n = 1..10350</a> (rows 1..500 of triangle, flattened)
%F T(2^m,1) = 2^m, for m >= 0. - _Paolo Xausa_, Jun 19 2022
%e Triangle begins:
%e [1];
%e [2];
%e [1, 2], [3];
%e [4];
%e [2, 3], [5];
%e [1, 2, 3], [6];
%e [3, 4], [7];
%e [8];
%e [2, 3, 4], [4, 5], [9];
%e [1, 2, 3, 4], [10];
%e [5, 6], [11];
%e [3, 4, 5], [12];
%e [6, 7], [13];
%e [2, 3, 4, 5], [14];
%e [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15];
%e [16];
%e [8, 9], [17];
%e [3, 4, 5, 6], [5, 6, 7], [18];
%e [9, 10], [19];
%e [2, 3, 4, 5, 6], [20];
%e [1, 2, 3, 4, 5, 6], [6, 7, 8], [10, 11], [21];
%e [4, 5, 6, 7], [22];
%e [11, 12], [23];
%e [7, 8, 9], [24];
%e [3, 4, 5, 6, 7], [12, 13], [25];
%e [5, 6, 7, 8], [26];
%e [2, 3, 4, 5, 6, 7], [8, 9, 10], [13, 14], [27];
%e [1, 2, 3, 4, 5, 6, 7], [28];
%e ...
%e For n = 9 there are three partitions of 9 into consecutive parts, they are [9], [5, 4], [4, 3, 2], so the 9th row of triangle is [2, 3, 4], [4, 5], [9].
%e Note that in the below diagram the number of horizontal line segments in the n-th row equals A001227(n), the number of partitions of n into consecutive parts, so we can find the partitions of n into consecutive parts as follows: consider the vertical blocks of numbers that start exactly in the n-th row of the diagram, for example: for n = 15 consider the vertical blocks of numbers that start exactly in the 15th row. They are [1, 2, 3, 4, 5], [4, 5, 6], [7, 8], [15], equaling the 15th row of the above triangle.
%e Row _
%e 1 |1|_
%e 2 |_ 2|_
%e 3 |1| 3|_
%e 4 |2|_ 4|_
%e 5 |_ 2| 5|_
%e 6 |1|3|_ 6|_
%e 7 |2| 3| 7|_
%e 8 |3|_ 4|_ 8|_
%e 9 |_ 2| 4| 9|_
%e 10 |1|3| 5|_ 10|_
%e 11 |2|4|_ 5| 11|_
%e 12 |3| 3| 6|_ 12|_
%e 13 |4|_ 4| 6| 13|_
%e 14 |_ 2|5|_ 7|_ 14|_
%e 15 |1|3| 4| 7| 15|_
%e 16 |2|4| 5| 8|_ 16|_
%e 17 |3|5|_ 6|_ 8| 17|_
%e 18 |4| 3| 5| 9|_ 18|_
%e 19 |5|_ 4| 6| 9| 19|_
%e 20 |_ 2|5| 7|_ 10|_ 20|_
%e 21 |1|3|6|_ 6| 10| 21|_
%e 22 |2|4| 4| 7| 11|_ 22|_
%e 23 |3|5| 5| 8|_ 11| 23|_
%e 24 |4|6|_ 6| 7| 12|_ 24|_
%e 25 |5| 3|7|_ 8| 12| 25|_
%e 26 |6|_ 4| 5| 9|_ 13|_ 26|_
%e 27 |_ 2|5| 6| 8| 13| 27|_
%e 28 |1|3|6| 7| 9| 14| 28|
%e ...
%e The diagram is infinite. For more information about the diagram see A286001.
%e For an amazing connection with sum of divisors function (A000203) see A237593.
%t Table[With[{h = Floor[n/2] - Boole[EvenQ@ n]},Append[Array[Which[Total@ # == n, #, Total@ Most@ # == n, Most[#], True, Nothing] &@ NestWhile[Append[#, #[[-1]] + 1] &, {#}, Total@ # <= n &, 1, h - # + 1] &, h], {n}]], {n, 24}] // Flatten (* _Michael De Vlieger_, Oct 22 2019 *)
%Y Mirror of A299765.
%Y Row n has length A204217(n).
%Y Row sums give A245579.
%Y Column 1 gives A118235.
%Y Right border gives A000027.
%Y Records give A000027.
%Y Where records occur gives A285899.
%Y The number of partitions into consecutive parts in row n is A001227(n).
%Y For tables of partitions into consecutive parts see A286000 and A286001.
%Y Cf. A000203, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774, A328361, A328362.
%K nonn,tabf
%O 1,2
%A _Omar E. Pol_, Oct 22 2019