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

1,1

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.

Links

Table of n, a(n) for n=1..40.

Index to sequences related to Olympiads.

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

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

Sequence in context: A220200 A242656 A264851 * A338485 A344334 A225882

Adjacent sequences: A345283 A345284 A345285 * A345287 A345288 A345289

Keyword

nonn

Author

Bernard Schott, Jun 13 2021

Status

approved