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%I #6 Aug 03 2019 21:47:10
%S 0,0,0,3,3,6,8,11,13,18,19,24,27,33,36,44,47,54,59,66,70,81,84,95,100,
%T 111,116,128,134,146,153,165,172,186,192,207,215,230,238,256,264,281,
%U 291,309,319,340,349,369,380,400,411,432,442,464,475,497,508,532
%N Number of squarefree parts in the partitions of n into 3 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (mu(i)^2 + mu(j)^2 + mu(n-i-j)^2), where mu is the Möbius function (A008683).
%e Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
%e 1+1+8
%e 1+1+7 1+2+7
%e 1+2+6 1+3+6
%e 1+1+6 1+3+5 1+4+5
%e 1+1+5 1+2+5 1+4+4 2+2+6
%e 1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
%e 1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
%e 1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
%e -----------------------------------------------------------------------
%e n | 3 4 5 6 7 8 9 10 ...
%e -----------------------------------------------------------------------
%e a(n) | 3 3 6 8 11 13 18 19 ...
%e -----------------------------------------------------------------------
%t Table[Sum[Sum[MoebiusMu[i]^2 + MoebiusMu[j]^2 + MoebiusMu[n - i - j]^2, {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 50}]
%Y Cf. A008683.
%K nonn
%O 0,4
%A _Wesley Ivan Hurt_, Aug 03 2019