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

%I A273140 #25 Dec 12 2016 08:59:58
%S A273140 1,3,6,11,17,25,34,46,59,74,90,109,129,151,174,201,229,259,290,323,
%T A273140 358,394,434,475,518,562,609,657,707,758,814,871,930,990,1052,1116,
%U A273140 1181,1249,1318,1389,1462,1536,1615,1695,1777,1860,1946,2033,2122,2212,2305,2400,2496,2594,2694,2795
%N A273140 Number of parts in the corner of size n X n of the modular table of partitions described in Comments.
%C A273140 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.
%e A273140 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.
%e A273140 .
%e A273140 .   Row   _ _ _ _       Parts
%e A273140 .    1   |_| | | |        4
%e A273140 .    2   |_ _| | |        3
%e A273140 .    3   |_ _ _| |        2
%e A273140 .    4   |_ _|   |        2
%e A273140 .                       ----
%e A273140 .                  Total 11
%e A273140 .
%e A273140 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.
%e A273140 .
%e A273140 .   Row   _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _       Parts
%e A273140 .    1   |_| | | | | | | | | | | | | | | | | | | |        20
%e A273140 .    2   |_ _| | | | | | | | | | | | | | | | | | |        19
%e A273140 .    3   |_ _ _| | | | | | | | | | | | | | | | | |        18
%e A273140 .    4   |_ _|   | | | | | | | | | | | | | | | | |        18
%e A273140 .    5   |_ _ _ _| | | | | | | | | | | | | | | | |        17
%e A273140 .    6   |_ _ _|   | | | | | | | | | | | | | | | |        17
%e A273140 .    7   |_ _ _ _ _| | | | | | | | | | | | | | | |        16
%e A273140 .    8   |_ _|   |   | | | | | | | | | | | | | | |        17
%e A273140 .    9   |_ _ _ _|   | | | | | | | | | | | | | | |        16
%e A273140 .   10   |_ _ _|     | | | | | | | | | | | | | | |        16
%e A273140 .   11   |_ _ _ _ _ _| | | | | | | | | | | | | | |        15
%e A273140 .   12   |_ _ _|   |   | | | | | | | | | | | | | |        16
%e A273140 .   13   |_ _ _ _ _|   | | | | | | | | | | | | | |        15
%e A273140 .   14   |_ _ _ _|     | | | | | | | | | | | | | |        15
%e A273140 .   15   |_ _ _ _ _ _ _| | | | | | | | | | | | | |        14
%e A273140 .   16   |_ _|   |   |   | | | | | | | | | | | | |        16
%e A273140 .   17   |_ _ _ _|   |   | | | | | | | | | | | | |        15
%e A273140 .   18   |_ _ _|     |   | | | | | | | | | | | | |        15
%e A273140 .   19   |_ _ _ _ _ _|   | | | | | | | | | | | | |        14
%e A273140 .   20   |_ _ _ _ _|     | | | | | | | | | | | | |        14
%e A273140 .                                                       -----
%e A273140 .                                                  Total 323
%e A273140 .
%Y A273140 Cf. A000041, A006128, A135010, A141285, A194446, A206437, A211992, A220482, A228525, A278602.
%K A273140 nonn
%O A273140 1,2
%A A273140 _Omar E. Pol_, May 16 2016