A288529 a(n) is the minimum number of rows from the table described in A286000 that are required to represent the partitions of n into consecutive parts.

1, 2, 4, 4, 6, 8, 8, 8, 11, 13, 12, 14, 14, 17, 19, 16, 18, 21, 20, 24, 26, 25, 24, 26, 29, 29, 32, 34, 30, 34, 32, 32, 38, 37, 41, 43, 38, 41, 44, 44, 42, 48, 44, 51, 53, 49, 48, 50, 55, 54, 56, 59, 54, 62, 64, 62, 62, 61, 60, 67, 62, 65, 71, 64, 74, 76, 68, 75, 74, 76, 72, 80, 74, 77, 84, 83, 87, 89, 80, 84, 89, 85

Offset: 1

Author

Omar E. Pol, Jun 19 2017

Keywords

• nonn

Comments

a(n) has the same definition related to the table A286001 which is another version of the table A286000.

First differs from A288772 at a(11), which shares infinitely many terms.

Examples

			Figures A..D show the evolution of the table of partitions into consecutive parts described in A286000, for n = 8..11:
.     ---------------------------------------------------------------------
Figure:      A            B                    C                  D
.     ---------------------------------------------------------------------
.    n:      8            9                   10                 11
Row   ---------------------------------------------------------------------
1     |  1;        |  1;             |   1;             |   1;            |
1     |  2;        |  2;             |   2;             |   2;            |
3     |  3,  2;    |  3,  2;         |   3,  2;         |   3,  2;        |
4     |  4,  1;    |  4,  1;         |   4,  1;         |   4,  1;        |
5     |  5,  3;    |  5,  3;         |   5,  3;         |   5,  3;        |
6     |  6,  2,  3;|  6,  2,  3;     |   6,  2,  3;     |   6,  2,  3;    |
7     |  7,  4,  2;|  7,  4,  2;     |   7,  4,  2;     |   7,  4,  2;    |
8     | [8], 3,  1;|  8,  3,  1;     |   8,  3,  1;     |   8,  3,  1;    |
9     |            | [9],[5],[4];    |   9,  5,  4;     |   9,  5,  4;    |
10    |            | 10, [4],[3],  4;| [10], 4,  3, [4];|  10,  4,  3;  4;|
11    |            | 11,  6, [2],  3;|  11,  6,  2; [3];| [11],[6], 2,  3;|
12    |            |                 |  12,  5,  5, [2];|  12, [5], 5,  2;|
13    |            |                 |  13,  7,  4, [1];|                 |
.     ---------------------------------------------------------------------
. a(n):      8              11                13                 12
.     ---------------------------------------------------------------------
For n = 8 we need a table with at least 8 rows, so a(8) = 8.
For n = 9 we need a table with at least 11 rows, so a(9) = 11.
For n = 10 we need a table with at least 13 rows, so a(10) = 13.
For n = 11 we need a table with at least 12 rows, so a(11) = 12.
		

Crossrefs

Cf. A109814, A237593, A286000, A286001, A288772, A288773, A288774.

Formula

a(n) = A109814(n) + n - 1.