A225596 Sum of largest parts of all partitions of n plus n. Also, total number of parts in all partitions of n plus n.

0, 2, 5, 9, 16, 25, 41, 61, 94, 137, 202, 286, 411, 569, 794, 1083, 1479, 1982, 2662, 3517, 4650, 6073, 7921, 10229, 13198, 16876, 21548, 27321, 34573, 43482, 54593, 68166, 84959, 105399, 130496, 160911, 198050, 242849, 297239, 362626, 441586, 536145

Offset: 0

Author

Omar E. Pol, Aug 01 2013

Keywords

• nonn

Comments

a(n) is also the number of horizontal toothpicks (or the total length of all horizontal boundary segments) in the diagram of regions of the set of partitions of n, see example. A093694(n) is the number of vertical toothpicks in the diagram. See also A225610. For a minimalist version of the diagram see A211978. For the definition of "region" see A206437.

Examples

			For n = 7 the sum of largest parts of all partitions of 7 plus 7 is (7+4+5+3+6+3+4+2+5+3+4+2+3+2+1) + 7 = 54 + 7 = 61. On the other hand the number of toothpicks in horizontal direction in the diagram of regions of the set of partitions of 7 is equal to 61, so a(7) = 61.
.
.                  Diagram of regions       Horizontal
Partitions         and partitions of 7      toothpicks
of 7
.                     _ _ _ _ _ _ _
7                    |_ _ _ _      |             7
4+3                  |_ _ _ _|_    |             4
5+2                  |_ _ _    |   |             5
3+2+2                |_ _ _|_ _|_  |             3
6+1                  |_ _ _      | |             6
3+3+1                |_ _ _|_    | |             3
4+2+1                |_ _    |   | |             4
2+2+2+1              |_ _|_ _|_  | |             2
5+1+1                |_ _ _    | | |             5
3+2+1+1              |_ _ _|_  | | |             3
4+1+1+1              |_ _    | | | |             4
2+2+1+1+1            |_ _|_  | | | |             2
3+1+1+1+1            |_ _  | | | | |             3
2+1+1+1+1+1          |_  | | | | | |             2
1+1+1+1+1+1+1        |_|_|_|_|_|_|_|             1
.                                                7
.                                              _____
.                                       Total   61
.
		

Crossrefs

Cf. A000041, A006128, A093694, A135010, A139250, A141285, A186114, A194446, A194447, A206437, A211978, A220517, A225600, A225610.

Formula

a(n) = A006128(n) + n = A225610(n) - A093694(n).

a(n) = n + sum_{k=1..A000041(n)} A141285(k), n >= 1.