A163986 Sum of all repeated parts of all partitions of n.

0, 0, 2, 4, 13, 26, 55, 92, 161, 253, 401, 595, 901, 1288, 1863, 2611, 3665, 5016, 6895, 9273, 12501, 16591, 22001, 28820, 37753, 48901, 63285, 81217, 104049, 132328, 168061, 212041, 267105, 334654, 418473, 520836, 647101, 800496, 988495, 1216138, 1493441, 1827822, 2233225, 2720138, 3307613, 4010941, 4855577, 5863345, 7069009, 8502628, 10211201

Offset

0,3

Comments

See A163985 for more information.

Links

Table of n, a(n) for n=0..50.

Omar E. Pol, Illustration of the shell model of partitions (2D view)

Omar E. Pol, Illustration of the shell model of partitions (3D view)

Formula

a(0)=0, a(n)=A066186(n)-A163985(n), for n>0.

Example

For n=4, the five partitions of 4 are {(4);(2,2);(3,1);(2,1,1);(1,1,1,1)}. Since 1 and 2 are repeated parts and 3 and 4 are not repeated parts (or isolated parts) then a(4)={(2+2)+(1)+(2+1+1)+(1+1+1+1)}=13.

Mathematica

Table[Total[Flatten[Select[Split[Sort[Flatten[IntegerPartitions[n]]]], Length[ #]>1&]]], {n, 0, 50}] (* Harvey P. Dale, Apr 30 2018 *)

Crossrefs

Cf. A000041, A066186, A163985.

Sequence in context: A360397 A018263 A018587 * A189583 A153936 A027301

Adjacent sequences: A163983 A163984 A163985 * A163987 A163988 A163989

Keyword

easy,nonn

Author

Omar E. Pol, Aug 14 2009

Extensions

More terms from Alois P. Heinz, Jan 30 2011

Status

approved