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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. 11
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 June 21 03:09 EDT 2021. Contains 345351 sequences. (Running on oeis4.)