OFFSET
1,10
FORMULA
a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (1 - ceiling(i*j*(n-i-j)/n) + floor(i*j*(n-i-j)/n)) * (1 - [i = j]) * (1 - [n-i = 2*j]) * (1 - [n-j = 2*i]), where [ ] is the Iverson bracket.
EXAMPLE
a(10) = 2; [1,4,5], [2,3,5], with all parts distinct;
a(12) = 3; [1,3,8], [2,4,6], [3,4,5], with all parts distinct.
MATHEMATICA
Table[Sum[Sum[(1 - KroneckerDelta[i, j]) (1 - KroneckerDelta[n - j, 2 i]) (1 - KroneckerDelta[n - i, 2 j]) (1 - Ceiling[i*j*(n - i - j)/n] + Floor[i*j*(n - i - j)/n]), {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, Apr 11 2021
STATUS
approved