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

0,2

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.

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

Table of n, a(n) for n=0..87.

The IMO compendium, Problem 4, 18th Austrian Mathematical Olympiad, 1987.

Index to sequences related to Olympiads.

Formula

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

Example

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.

Crossrefs

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

Sequence in context: A245525 A233412 A327892 * A278266 A088200 A073103

Adjacent sequences: A339374 A339375 A339376 * A339378 A339379 A339380

Keyword

nonn

Author

Bernard Schott, Dec 02 2020

Status

approved