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

1,1

Links

Table of n, a(n) for n=1..40.

Index entries for related partition-counting sequences

Formula

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

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

Example

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

Mathematica

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

Crossrefs

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

Sequence in context: A366967 A360096 A016436 * A318180 A301115 A050843

Adjacent sequences: A338966 A338967 A338968 * A338970 A338971 A338972

Keyword

nonn

Author

Stefano Spezia, Dec 23 2020

Status

approved