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A335221
Number of Heronian triangles with perimeter A330869(n) whose side lengths are squarefree.
0
1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 3, 1, 2, 2, 1, 4, 1, 1, 1, 4, 3, 1, 2, 2, 1, 2, 3, 1, 1, 1, 3, 1, 1, 2, 1, 2, 1, 5, 2, 1, 1, 4, 2, 3, 3, 1, 1, 1, 1, 1, 2, 4, 1, 3, 1, 2, 1, 3, 9, 1, 1, 6, 1, 1, 1, 3, 1, 1, 1, 6, 6, 2, 1, 2, 2, 2, 2, 1, 1, 2, 1, 1, 5, 7, 1, 3, 4, 4, 3, 1, 2, 3, 2
OFFSET
1,5
LINKS
Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Integer Triangle
FORMULA
a(n) = Sum_{k=1..floor(b(n)/3)} Sum_{i=k..floor((b(n)-k)/2)} sign(floor((i+k)/(b(n)-i-k+1))) * (1-ceiling(c)+floor(c)) * mu(k)^2 * mu(i)^2 * mu(b(n)-i-k)^2, where c = sqrt((b(n)/2)*(b(n)/2-i)*(b(n)/2-k)*(b(n)/2-(b(n)-i-k))), mu is the Möbius function (A008683) and b(n) = A330869(n).
EXAMPLE
a(1) = 1; there is one Heronian triangle with perimeter A330869(1) = 16 whose side lengths are squarefree, which is [5,5,6].
a(2) = 1; there is one Heronian triangle with perimeter A330869(2) = 36 whose side lengths are squarefree, [10,13,13].
CROSSREFS
Sequence in context: A240061 A162320 A241911 * A136610 A377365 A326371
KEYWORD
nonn
AUTHOR
Wesley Ivan Hurt, May 27 2020
STATUS
approved