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.

4, 11, 21, 41, 67, 118, 181, 292, 437, 664, 958, 1412, 1983, 2819, 3899, 5406, 7328, 9977, 13317, 17817, 23497, 30967, 40349, 52573, 67784, 87320, 111601, 142395, 180432, 228317, 287110, 360476, 450261, 561346, 696699, 863199, 1065055, 1311824, 1610026, 1972444

Offset: 1

Author

Stefano Spezia, Dec 23 2020

Keywords

• nonn

Examples

			Illustrations for n = 1..6:
      _           _ _          _ _ _
     |_|         |_ _|        |_ _ _|
                 |_|_|        |_ _|_|
                              |_|_|_|
  a(1) = 4     a(2) = 11     a(3) = 21
   _ _ _ _     _ _ _ _ _    _ _ _ _ _ _
  |_ _ _ _|   |_ _ _ _ _|  |_ _ _ _ _ _|
  |_ _ _|_|   |_ _ _ _|_|  |_ _ _ _ _|_|
  |_ _|_ _|   |_ _ _|_ _|  |_ _ _ _|_ _|
  |_ _|_|_|   |_ _ _|_|_|  |_ _ _ _|_|_|
  |_|_|_|_|   |_ _|_ _|_|  |_ _ _|_ _ _|
              |_ _|_|_|_|  |_ _ _|_ _|_|
              |_|_|_|_|_|  |_ _ _|_|_|_|
                           |_ _|_ _|_ _|
                           |_ _|_ _|_|_|
                           |_ _|_|_|_|_|
                           |_|_|_|_|_|_|
  a(4) = 41    a(5) = 67    a(6) = 118
		

Links

• Index entries for related partition-counting sequences

Crossrefs

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

Programs

Mathematica

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

Formula

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

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