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

%I A002220 M3395 N1374 #43 Apr 21 2024 23:50:54
%S A002220 1,4,10,30,65,173,343,778,1518,3088,5609,10959,18990,34441,58903,
%T A002220 102044,167499,282519,451529,737208,1160102,1836910,2828466,4410990,
%U A002220 6670202,10161240,15186315,22758131,33480869
%N A002220 a(n) is the number of partitions of 3n that can be obtained by adding together three (not necessarily distinct) partitions of n.
%D A002220 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002220 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002220 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 A002220 From _Gus Wiseman_, Apr 20 2024: (Start)
%e A002220 The a(1) = 1 through a(3) = 10 triquanimous partitions:
%e A002220   (111)  (222)     (333)
%e A002220          (2211)    (3321)
%e A002220          (21111)   (32211)
%e A002220          (111111)  (33111)
%e A002220                    (222111)
%e A002220                    (321111)
%e A002220                    (2211111)
%e A002220                    (3111111)
%e A002220                    (21111111)
%e A002220                    (111111111)
%e A002220 (End)
%Y A002220 See A002219 for further details. Cf. A002221, A002222, A213074.
%Y A002220 A column of A213086.
%Y A002220 For biquanimous we have A002219, ranks A357976.
%Y A002220 For non-biquanimous we have A371795, ranks A371731, even case A006827.
%Y A002220 The Heinz numbers of these partitions are given by A371955.
%Y A002220 The strict case is A372122.
%Y A002220 A321451 counts non-quanimous partitions, ranks A321453.
%Y A002220 A321452 counts quanimous partitions, ranks A321454.
%Y A002220 A371783 counts k-quanimous partitions.
%Y A002220 Cf. A035470, A064914, A237258, A321142, A371737, A371792, A371796.
%K A002220 nonn
%O A002220 1,2
%A A002220 _N. J. A. Sloane_
%E A002220 Edited by _N. J. A. Sloane_, Jun 03 2012
%E A002220 a(12)-a(20) from _Alois P. Heinz_, Jul 10 2012
%E A002220 a(21)-a(29) from _Sean A. Irvine_, Sep 05 2013