A339377 Number of triples (x, y, z) of natural numbers satisfying x+y = n and 2*x*y = z^2.

1, 2, 2, 4, 2, 2, 4, 2, 2, 6, 2, 4, 4, 2, 2, 4, 2, 4, 6, 4, 2, 4, 4, 2, 4, 2, 2, 8, 2, 2, 4, 2, 2, 10, 4, 2, 6, 2, 4, 4, 2, 4, 4, 4, 4, 6, 2, 2, 4, 2, 2, 10, 2, 2, 8, 4, 2, 10, 2, 4, 4, 2, 2, 6, 2, 2, 10, 4, 4, 4, 2, 2, 6, 4, 2, 4, 4, 4, 4, 2, 2, 10, 4, 4, 4, 4, 4, 4

Offset: 0

Author

Bernard Schott, Dec 02 2020

Keywords

• nonn

Comments

This sequence is inspired by the 4th problem proposed during the second day of the final round of the 18th Austrian Mathematical Olympiad in 1987. The problem asked to find all triples solutions (x, y, z) only for n = 1987 (see Link, Reference and last example).

Some properties:

-> Inequalities, 0 <= x, y <= n; 0 <= z <= floor(n*sqrt(2)/2)

-> z is even and (x,y) are not together even.

-> a(n) = 1 iff n = 0, and the only solution is (0,0,0).

-> for n >= 1, a(n) >= 2 because (0,n,0) and (n,0,0) are always solutions.

-> a(n) is even for n >= 1.

-> If n = 3k, then (k,2k,2k) and (2k,k,2k) are solutions.

-> If 2*(n-1) = m^2, then (1,n-1,m) and (n-1,1,m) are solutions (with n in A058331).

-> The formula for n>0 comes from (x+y=n and 2*x*y=z^2) <==> n^2 = |x-y|^2 + 2*z^2.

Examples

			a(9) = 6 and these 6 solutions are: (0, 9, 0), (1, 8, 4), (3, 6, 6), (6, 3, 6), (8, 1, 4), (9, 0, 0).
a(1987) = 4 and these 4 solutions are: (0, 1987, 0), (529, 1458, 1242), (1458, 529, 1242), (1987, 0, 0); this is the answer to the Olympiad problem in link.
		

References

• Steve Dinh, The Hard Mathematical Olympiad Problems And Their Solutions, AuthorHouse, 2011, Problem 4 of Austrian Mathematical Olympiad 1987, page 29 [Warning: solution proposed in this book has a mistake with (x, y, z) = ([0, 1987], 1987-x, sqrt(2xy))].

Links

• The IMO compendium, Problem 4, 18th Austrian Mathematical Olympiad, 1987.
• Index to sequences related to Olympiads.

Crossrefs

Cf. A058331, A218799, A339378 (variant with x+y = n and x*y = z^2).

Formula

a(0)=A218799(0); then for n>=1, a(n)=2*A218799(n) (remark from Hugo Pfoertner, Dec 02 2020).