A345286 a(n) is the number of large or small squares that are used to tile primary squares of type 1 (see A344331) whose side length is A345285(n).

20, 90, 272, 468, 650, 1280, 1332, 2900, 3600, 2450, 7650, 5760, 4160, 6642, 10388, 810, 16400, 10100, 1088, 25578, 29952, 14762, 27540, 20880, 42048, 50960, 54900, 41600, 28730, 65610, 81920, 90650, 60500, 38612, 98100, 50850, 125712, 85248, 142400, 149940

Offset: 1

Author

Bernard Schott, Jun 13 2021

Keywords

• nonn

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.

Examples

			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.
		

References

• Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.

Links

• Index to sequences related to Olympiads.

Crossrefs

Cf. A071253, A344330, A344331, A344333, A344334, A345285, A345287.