A328365 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.
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, 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, 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
Offset: 1
Examples
Triangle begins: [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, 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, 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]; [3, 4, 5, 6, 7], [12, 13], [25]; [5, 6, 7, 8], [26]; [2, 3, 4, 5, 6, 7], [8, 9, 10], [13, 14], [27]; [1, 2, 3, 4, 5, 6, 7], [28]; ... 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]. 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. Row _ 1 |1|_ 2 |_ 2|_ 3 |1| 3|_ 4 |2|_ 4|_ 5 |_ 2| 5|_ 6 |1|3|_ 6|_ 7 |2| 3| 7|_ 8 |3|_ 4|_ 8|_ 9 |_ 2| 4| 9|_ 10 |1|3| 5|_ 10|_ 11 |2|4|_ 5| 11|_ 12 |3| 3| 6|_ 12|_ 13 |4|_ 4| 6| 13|_ 14 |_ 2|5|_ 7|_ 14|_ 15 |1|3| 4| 7| 15|_ 16 |2|4| 5| 8|_ 16|_ 17 |3|5|_ 6|_ 8| 17|_ 18 |4| 3| 5| 9|_ 18|_ 19 |5|_ 4| 6| 9| 19|_ 20 |_ 2|5| 7|_ 10|_ 20|_ 21 |1|3|6|_ 6| 10| 21|_ 22 |2|4| 4| 7| 11|_ 22|_ 23 |3|5| 5| 8|_ 11| 23|_ 24 |4|6|_ 6| 7| 12|_ 24|_ 25 |5| 3|7|_ 8| 12| 25|_ 26 |6|_ 4| 5| 9|_ 13|_ 26|_ 27 |_ 2|5| 6| 8| 13| 27|_ 28 |1|3|6| 7| 9| 14| 28| ... The diagram is infinite. For more information about the diagram see A286001. For an amazing connection with sum of divisors function (A000203) see A237593.
Links
- Paolo Xausa, Table of n, a(n) for n = 1..10350 (rows 1..500 of triangle, flattened)
Crossrefs
Programs
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Mathematica
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 *)
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