A344334 a(n) is the number of large or small squares that are used to tile primitive squares of type 1 whose length of side is A344333(n).

20, 90, 272, 468, 650, 1332, 2900, 3600, 2450, 7650, 4160, 6642, 10388, 16400, 10100, 25578, 14762, 27540, 20880, 42048, 50960, 54900, 28730, 90650, 60500, 38612, 98100, 50850, 125712, 142400, 149940, 65792, 141570, 116948, 214650, 83810, 105300, 265232, 354368

Offset: 1

Author

Bernard Schott, Jun 02 2021

Keywords

• nonn

Comments

Some notations: s = side of the 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) with gcd(a, b) = 1.

Every primitive 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(9) = 2450 < a(8) = 3600.

Every term is even.

If a = 1 and b = n > 1, then number of squares z = n^2 * (n^2+1) is in A071253 \ {0,2}.

Examples

			Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
		

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. A344330, A344331, A344332, A344333.

Cf. A071253 \ {0,2} is a subsequence.