A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.

1, 3, 6, 11, 17, 25, 34, 46, 59, 74, 90, 109, 129, 151, 174, 201, 229, 259, 290, 323, 358, 394, 434, 475, 518, 562, 609, 657, 707, 758, 814, 871, 930, 990, 1052, 1116, 1181, 1249, 1318, 1389, 1462, 1536, 1615, 1695, 1777, 1860, 1946, 2033, 2122, 2212, 2305, 2400, 2496, 2594, 2694, 2795

Offset: 1

Author

Omar E. Pol, May 16 2016

Keywords

• nonn

Comments

Consider an infinite dissection of the fourth quadrant of the square grid in which, apart from the axes x and y, the k-th horizontal line segment has length A141285(n) and the n-th vertical line segment has length A194446(n). Both line segments shares the point (A141285(n),n). Note that in the infinite table there are no partitions because every row contains an infinite number of parts. On the other hand, taking only the first k sections from the table we have a representation of the partitions of k. For the definition of "region" see A206437. For the definition of "section" see A135010.

Examples

			For n = 4 the corner of size 4 X 4 of the modular table of partitions contains 11 parts as shown below, so a(4) = 11.
.
.   Row   _ _ _ _       Parts
.    1   |_| | | |        4
.    2   |_ _| | |        3
.    3   |_ _ _| |        2
.    4   |_ _|   |        2
.                       ----
.                  Total 11
.
For n = 20 the corner of size 20 X 20 of the modular table of partitions contains 323 parts as shown below, so a(20) = 323.
.
.   Row   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _       Parts
.    1   |_| | | | | | | | | | | | | | | | | | | |        20
.    2   |_ _| | | | | | | | | | | | | | | | | | |        19
.    3   |_ _ _| | | | | | | | | | | | | | | | | |        18
.    4   |_ _|   | | | | | | | | | | | | | | | | |        18
.    5   |_ _ _ _| | | | | | | | | | | | | | | | |        17
.    6   |_ _ _|   | | | | | | | | | | | | | | | |        17
.    7   |_ _ _ _ _| | | | | | | | | | | | | | | |        16
.    8   |_ _|   |   | | | | | | | | | | | | | | |        17
.    9   |_ _ _ _|   | | | | | | | | | | | | | | |        16
.   10   |_ _ _|     | | | | | | | | | | | | | | |        16
.   11   |_ _ _ _ _ _| | | | | | | | | | | | | | |        15
.   12   |_ _ _|   |   | | | | | | | | | | | | | |        16
.   13   |_ _ _ _ _|   | | | | | | | | | | | | | |        15
.   14   |_ _ _ _|     | | | | | | | | | | | | | |        15
.   15   |_ _ _ _ _ _ _| | | | | | | | | | | | | |        14
.   16   |_ _|   |   |   | | | | | | | | | | | | |        16
.   17   |_ _ _ _|   |   | | | | | | | | | | | | |        15
.   18   |_ _ _|     |   | | | | | | | | | | | | |        15
.   19   |_ _ _ _ _ _|   | | | | | | | | | | | | |        14
.   20   |_ _ _ _ _|     | | | | | | | | | | | | |        14
.                                                       -----
.                                                  Total 323
.
		

Crossrefs

Cf. A000041, A006128, A135010, A141285, A194446, A206437, A211992, A220482, A228525, A278602.