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Number of integer partitions (w,x,y) summing to n such that 2w = 3x + 4y.
4

%I #8 Nov 02 2022 11:52:59

%S 0,0,0,0,0,0,0,0,1,0,0,1,0,1,0,0,1,0,1,1,0,1,1,1,1,0,1,1,1,1,1,1,1,2,

%T 1,1,1,1,2,1,1,2,1,2,2,1,2,1,2,2,1,2,2,2,2,2,2,2,2,2,2,2,2,3,2,2,3,2,

%U 3,2,2,3,2,3,3,2,3,3,3,3,2,3,3,3,3,3,3

%N Number of integer partitions (w,x,y) summing to n such that 2w = 3x + 4y.

%e The partitions for n = 34, 64, 89, 119, 144:

%e (21,10,3) (39,22,3) (54,32,3) (72,44,3) (87,54,3)

%e (40,16,8) (55,26,8) (73,38,8) (88,48,8)

%e (56,20,13) (74,32,13) (89,42,13)

%e (75,26,18) (90,36,18)

%e (91,30,23)

%t Table[Length[Select[IntegerPartitions[n,{3}],2*#[[1]]==3*#[[2]]+4*#[[3]]&]],{n,0,100}]

%o (Python)

%o def A357849(n): return sum(1 for y in range(1,n-1) if (m:=2*n-6*y)>=5*y and 5*(n-y)>=2*m and m%5==0) # _Chai Wah Wu_, Nov 02 2022

%Y Partitions are counted by A000041, strict A000009.

%Y The ordered version appears to be A008676, ranked by A357489.

%Y These partitions are ranked by A358102.

%K nonn

%O 0,34

%A _Gus Wiseman_, Nov 02 2022