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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.
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%I #42 Nov 13 2021 04:45:02

%S 4,11,21,41,67,118,181,292,437,664,958,1412,1983,2819,3899,5406,7328,

%T 9977,13317,17817,23497,30967,40349,52573,67784,87320,111601,142395,

%U 180432,228317,287110,360476,450261,561346,696699,863199,1065055,1311824,1610026,1972444

%N 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 <a href="/index/Par#partitions">Index entries for related partition-counting sequences</a>

%F a(n) = (n + 1)*A000041(n) + n + A006128(n).

%F a(n) = A066186(n) + A000041(n) + A225596(n).

%e Illustrations for n = 1..6:

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%e a(1) = 4 a(2) = 11 a(3) = 21

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%e a(4) = 41 a(5) = 67 a(6) = 118

%t a[n_]:=(n+1)PartitionsP[n]+n+Sum[DivisorSigma[0,m] PartitionsP[n-m], {m,n}]; Table[a[n],{n,40}]]

%Y Cf. A000041, A006128, A066186, A211992, A225596, A278355.

%K nonn

%O 1,1

%A _Stefano Spezia_, Dec 23 2020