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%I #10 Jun 18 2020 19:38:48
%S 0,0,0,0,0,0,0,1,1,3,2,5,5,8,7,12,11,16,15,21,20,27,25,33,32,40,38,48,
%T 46,56,54,65,63,75,72,85,83,96,93,108,105,120,117,133,130,147,143,161,
%U 158,176,172,192,188,208,204,225,221,243,238,261,257,280,275
%N Number of length-3 strict integer partitions of n such that the largest part is not the sum of the other two.
%F Conjectures from _Colin Barker_, May 15 2019: (Start)
%F G.f.: x^7*(1 + x + 2*x^2) / ((1 - x)^3*(1 + x)^2*(1 + x^2)*(1 + x + x^2)).
%F a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-5) - a(n-6) - a(n-7) + a(n-9) for n>9.
%F (End)
%F a(n) = A325696(n)/6. - _Alois P. Heinz_, Jun 18 2020
%e The a(7) = 1 through a(15) = 12 partitions (A = 10, B = 11, C = 12):
%e (421) (521) (432) (631) (542) (543) (643) (653) (654)
%e (531) (721) (632) (732) (652) (842) (753)
%e (621) (641) (741) (742) (851) (762)
%e (731) (831) (751) (932) (843)
%e (821) (921) (832) (941) (852)
%e (841) (A31) (861)
%e (931) (B21) (942)
%e (A21) (951)
%e (A32)
%e (A41)
%e (B31)
%e (C21)
%t Table[Length[Select[IntegerPartitions[n,{3}],UnsameQ@@#&&#[[1]]!=#[[2]]+#[[3]]&]],{n,0,30}]
%Y Cf. A000041, A001399, A005044, A008642, A069905, A124278.
%Y Cf. A325686, A325690, A325691, A325694, A325696.
%K nonn
%O 0,10
%A _Gus Wiseman_, May 15 2019
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