A344334 - OEIS

%I #12 Jun 02 2021 22:43:29

%S 20,90,272,468,650,1332,2900,3600,2450,7650,4160,6642,10388,16400,

%T 10100,25578,14762,27540,20880,42048,50960,54900,28730,90650,60500,

%U 38612,98100,50850,125712,142400,149940,65792,141570,116948,214650,83810,105300,265232,354368

%N 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).

%C 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.

%C Every term is of the form z = (a*b)^2 * (a^2+b^2) with gcd(a, b) = 1.

%C 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.

%C This sequence is not increasing: a(9) = 2450 < a(8) = 3600.

%C Every term is even.

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

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

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.

%e Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.

%e       ___ ___ _ ___ ___ _

%e      |   |   |_|   |   |_|

%e      |___|___|_|___|___|_|

%e      |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles

%e      |___|___|_|___|___|_|

%e      |   |   |_|   |   |_|              ___ ___ _

%e      |___|___|_|___|___|_|             |   |   |_|

%e      |   |   |_|   |   |_|             |___|___|_|

%e      |___|___|_|___|___|_|

%e      |   |   |_|   |   |_|

%e      |___|___|_|___|___|_|

%Y Cf. A344330, A344331, A344332, A344333.

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

%K nonn

%O 1,1

%A _Bernard Schott_, Jun 02 2021