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A328365 Irregular triangle read by rows in which row n lists in reverse order the partitions of n into consecutive parts. 9
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..99.

EXAMPLE

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.

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 *)

CROSSREFS

Mirror of A299765.

Row n has length A204217(n).

Row sums give A245579.

Column 1 gives A118235.

Right border gives A000027.

Records give A000027.

Where records occur gives A285899.

The number of partitions into consecutive parts in row n is A001227(n).

For tables of partitions into consecutive parts see A286000 and A286001.

Cf. A000203, A026792, A235791, A237048, A237591, A237593, A245092, A285914, A286013, A288529, A288772, A288773, A288774, A328361, A328362.

Sequence in context: A205123 A108715 A119671 * A033787 A165073 A176846

Adjacent sequences:  A328362 A328363 A328364 * A328366 A328367 A328368

KEYWORD

nonn,tabl

AUTHOR

Omar E. Pol, Oct 22 2019

STATUS

approved

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Last modified April 4 08:58 EDT 2020. Contains 333213 sequences. (Running on oeis4.)