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

1,1

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}.

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..39.

Index to sequences related to Olympiads.

Example

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

Crossrefs

Cf. A344330, A344331, A344332, A344333.

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

Sequence in context: A264851 A345286 A338485 * A225882 A281768 A225892

Adjacent sequences: A344331 A344332 A344333 * A344335 A344336 A344337

Keyword

nonn

Author

Bernard Schott, Jun 02 2021

Status

approved