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A338969 a(n) is the sum of the lengths of all the segments used to draw a rectangle of height partition(n) and width n divided into partition(n) rectangles of unit height, in turn, divided into rectangles of unit height and lengths corresponding to the parts of the partitions of n.

%I A338969 #45 Jun 20 2025 08:12:06
%S A338969 4,11,21,41,67,118,181,292,437,664,958,1412,1983,2819,3899,5406,7328,
%T A338969 9977,13317,17817,23497,30967,40349,52573,67784,87320,111601,142395,
%U A338969 180432,228317,287110,360476,450261,561346,696699,863199,1065055,1311824,1610026,1972444
%N A338969 a(n) is the sum of the lengths of all the segments used to draw a rectangle of height partition(n) and width n divided into partition(n) rectangles of unit height, in turn, divided into rectangles of unit height and lengths corresponding to the parts of the partitions of n.
%H A338969 <a href="/index/Par#partitions">Index entries for related partition-counting sequences</a>
%F A338969 a(n) = (n + 1)*A000041(n) + n + A006128(n).
%F A338969 a(n) = A066186(n) + A000041(n) + A225596(n).
%e A338969 Illustrations for n = 1..6:
%e A338969       _           _ _          _ _ _
%e A338969      |_|         |_ _|        |_ _ _|
%e A338969                  |_|_|        |_ _|_|
%e A338969                               |_|_|_|
%e A338969   a(1) = 4     a(2) = 11     a(3) = 21
%e A338969    _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
%e A338969   |_ _ _ _|   |_ _ _ _ _|  |_ _ _ _ _ _|
%e A338969   |_ _ _|_|   |_ _ _ _|_|  |_ _ _ _ _|_|
%e A338969   |_ _|_ _|   |_ _ _|_ _|  |_ _ _ _|_ _|
%e A338969   |_ _|_|_|   |_ _ _|_|_|  |_ _ _ _|_|_|
%e A338969   |_|_|_|_|   |_ _|_ _|_|  |_ _ _|_ _ _|
%e A338969               |_ _|_|_|_|  |_ _ _|_ _|_|
%e A338969               |_|_|_|_|_|  |_ _ _|_|_|_|
%e A338969                            |_ _|_ _|_ _|
%e A338969                            |_ _|_ _|_|_|
%e A338969                            |_ _|_|_|_|_|
%e A338969                            |_|_|_|_|_|_|
%e A338969   a(4) = 41    a(5) = 67    a(6) = 118
%t A338969 a[n_]:=(n+1)PartitionsP[n]+n+Sum[DivisorSigma[0,m] PartitionsP[n-m], {m,n}]; Table[a[n],{n,40}]
%Y A338969 Cf. A000041, A006128, A066186, A211992, A225596, A278355.
%K A338969 nonn
%O A338969 1,1
%A A338969 _Stefano Spezia_, Dec 23 2020