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 z = (a*b)^2 * (a^2+b^2) = a*b*s with a < b.
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.
This sequence is not increasing: a(10) = 2450 < a(9) = 3600.
If gcd(a, b) = 1, then number of squares z = a*b * (a^2+b^2) is in A344334.
If a = 1 and b = n > 1, then number of squares z = n^2 * (n^2+1) is in A071253 \ {0,2}.
Every term is even.
REFERENCES
Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
EXAMPLE
The primary square with side A345285(1) = 10 can be tiled with a(1) = 20 small squares of side a = 1 and 20 large squares of side b = 2.
___ ___ _ ___ ___ _
| | |_| | |_|
|___|___|_|___|___|_|
| | |_| | |_| with 10 elementary 2 x 5 rectangles
|___|___|_|___|___|_|
| | |_| | |_| ___ ___ _
|___|___|_|___|___|_| | | |_|
| | |_| | |_| |___|___|_|
|___|___|_|___|___|_|
| | |_| | |_|
|___|___|_|___|___|_|
The primary square with side A345285(6) = 160 can be tiled with a(6) = 1280 small squares of side a = 2 and 1280 large squares of side b = 4.
CROSSREFS
KEYWORD
nonn
AUTHOR
Bernard Schott, Jun 13 2021
STATUS
approved