%I #17 Jun 06 2021 09:00:56

%S 0,0,0,1,1,2,2,3,3,5,4,5,5,7,7,9,8,11,11,13,11,15,14,18,15,20,19,23,

%T 20,24,24,27,24,30,29,34,30,37,36,42,36,45,44,50,44,54,54,59,52,62,63,

%U 68,57,69,70,78,65,78,78,88,74,86,87,98,84,98,98,107,93,109,108,120,102,124,123

%N Number of partitions of n into 3 squarefree parts.

%H Alois P. Heinz, <a href="/A307815/b307815.txt">Table of n, a(n) for n = 0..10000</a> (first 1001 terms from Rémy Sigrist)

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = [x^n y^3] Product_{k>=1} 1/(1 - mu(k)^2*y*x^k).

%F a(n) = Sum_{k=1..floor(n/3)} Sum_{i=k..floor((n-k)/2)} mu(i)^2 * mu(k)^2 * mu(n-i-k)^2, where mu is the Mobius function. - _Wesley Ivan Hurt_, May 09 2019

%e a(10) = 4 because we have [7, 2, 1], [6, 3, 1], [6, 2, 2] and [5, 3, 2].

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0$3], `if`(i<1, 0, b(n, i-1)+

%p `if`(numtheory[issqrfree](i), [0, b(n-i, min(i, n-i))[1..3][]], 0)))

%p end:

%p a:= n-> b(n$2)[4]:

%p seq(a(n), n=0..80); # _Alois P. Heinz_, Apr 30 2019

%t Array[Count[IntegerPartitions[#, {3}], _?(AllTrue[#, SquareFreeQ] &)] &, 75, 0]

%t (* Second program: *)

%t b[n_, i_] := b[n, i] = If[n == 0, {1, 0, 0, 0}, If[i < 1, {0, 0, 0, 0}, b[n, i - 1] + If[SquareFreeQ[i], {0, Sequence @@ b[n - i, Min[i, n - i]][[1 ;; 3]]}, {0, 0, 0, 0}]]];

%t a[n_] := b[n, n][[4]];

%t a /@ Range[0, 80] (* _Jean-François Alcover_, Jun 06 2021, after _Alois P. Heinz_ *)

%Y Cf. A005117, A008683, A068307, A071068, A073576, A098235, A280210, A307727.

%K nonn

%O 0,6

%A _Ilya Gutkovskiy_, Apr 30 2019