A002220 a(n) is the number of partitions of 3n that can be obtained by adding together three (not necessarily distinct) partitions of n. (Formerly M3395 N1374)

1, 4, 10, 30, 65, 173, 343, 778, 1518, 3088, 5609, 10959, 18990, 34441, 58903, 102044, 167499, 282519, 451529, 737208, 1160102, 1836910, 2828466, 4410990, 6670202, 10161240, 15186315, 22758131, 33480869, 49539505, 71952804, 104927068, 150968293, 217320618, 309411494

Offset

1,2

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Bert Dobbelaere, Table of n, a(n) for n = 1..56

N. Metropolis and P. R. Stein, An elementary solution to a problem in restricted partitions, J. Combin. Theory, 9 (1970), 365-376.

Example

From Gus Wiseman, Apr 20 2024: (Start)
The a(1) = 1 through a(3) = 10 triquanimous partitions:
  (111)  (222)     (333)
         (2211)    (3321)
         (21111)   (32211)
         (111111)  (33111)
                   (222111)
                   (321111)
                   (2211111)
                   (3111111)
                   (21111111)
                   (111111111)
(End)

Crossrefs

See A002219 for further details. Cf. A002221, A002222, A213074.

A column of A213086.

For biquanimous we have A002219, ranks A357976.

For non-biquanimous we have A371795, ranks A371731, even case A006827.

The Heinz numbers of these partitions are given by A371955.

The strict case is A372122.

A321451 counts non-quanimous partitions, ranks A321453.

A321452 counts quanimous partitions, ranks A321454.

A371783 counts k-quanimous partitions.

Cf. A035470, A064914, A237258, A321142, A371737, A371792, A371796.

Sequence in context: A330529 A048044 A047188 * A222807 A090578 A007713

Adjacent sequences: A002217 A002218 A002219 * A002221 A002222 A002223

Keyword

nonn,changed

Author

N. J. A. Sloane

Extensions

Edited by N. J. A. Sloane, Jun 03 2012

a(12)-a(20) from Alois P. Heinz, Jul 10 2012

a(21)-a(29) from Sean A. Irvine, Sep 05 2013

More terms from Bert Dobbelaere, Apr 28 2026

Status

approved