A002220 - OEIS

%I M3395 N1374 #43 Apr 21 2024 23:50:54

%S 1,4,10,30,65,173,343,778,1518,3088,5609,10959,18990,34441,58903,

%T 102044,167499,282519,451529,737208,1160102,1836910,2828466,4410990,

%U 6670202,10161240,15186315,22758131,33480869

%N a(n) is the number of partitions of 3n that can be obtained by adding together three (not necessarily distinct) partitions of n.

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

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

%H N. Metropolis and P. R. Stein, <a href="http://dx.doi.org/10.1016/S0021-9800(70)80091-6">An elementary solution to a problem in restricted partitions</a>, J. Combin. Theory, 9 (1970), 365-376.

%e From _Gus Wiseman_, Apr 20 2024: (Start)

%e The a(1) = 1 through a(3) = 10 triquanimous partitions:

%e   (111)  (222)     (333)

%e          (2211)    (3321)

%e          (21111)   (32211)

%e          (111111)  (33111)

%e                    (222111)

%e                    (321111)

%e                    (2211111)

%e                    (3111111)

%e                    (21111111)

%e                    (111111111)

%e (End)

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

%Y A column of A213086.

%Y For biquanimous we have A002219, ranks A357976.

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

%Y The Heinz numbers of these partitions are given by A371955.

%Y The strict case is A372122.

%Y A321451 counts non-quanimous partitions, ranks A321453.

%Y A321452 counts quanimous partitions, ranks A321454.

%Y A371783 counts k-quanimous partitions.

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

%K nonn

%O 1,2

%A _N. J. A. Sloane_

%E Edited by _N. J. A. Sloane_, Jun 03 2012

%E a(12)-a(20) from _Alois P. Heinz_, Jul 10 2012

%E a(21)-a(29) from _Sean A. Irvine_, Sep 05 2013