OFFSET
1,1
COMMENTS
Notation: s = side of the primary tiled squares, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
Every term is of the form s = a*b * (a^2+b^2), with 1 <= a < b, and corresponding z = (a*b)^2 * (a^2+b^2) (A345286).
Every such primary square is composed of m = a*b * (a^2+b^2) elementary rectangles of length L = a^2+b^2 and width W = a*b, so with an area A = a*b * (a^2+b^2) = m.
If gcd(a, b) = 1, then primitive sides of square s = a*b * (a^2+b^2) are in A344333 that is a subsequence.
If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.
If q is a term and integer r > 1, then q * r^4 is another term.
Every term is even.
REFERENCES
Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
EXAMPLE
a(1) = 10 and the primary square 10 X 10 can be tiled with A345286(1) = 20 small squares with side a = 1 and 20 large squares with side b = 2.
___ ___ _ ___ ___ _
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_| with 10 elementary 2 X 5 rectangles
|___|___|_|___|___|_|
| | |_| | |_| ___ ___ _
|___|___|_|___|___|_| | | |_|
| | |_| | |_| |___|___|_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
a(6) = 160 is the first side of an primary square that is not primitive and it corresponds to (a,b) = (2,4); the square 160 X 160 can be tiled with A345286(6) = 1280 small squares with side a = 2 and 1280 large squares with side b = 4.
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Jun 13 2021
STATUS
approved