A345286 - OEIS

%I #26 Dec 24 2021 12:45:37

%S 20,90,272,468,650,1280,1332,2900,3600,2450,7650,5760,4160,6642,10388,

%T 810,16400,10100,1088,25578,29952,14762,27540,20880,42048,50960,54900,

%U 41600,28730,65610,81920,90650,60500,38612,98100,50850,125712,85248,142400,149940

%N 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).

%C 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.

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

%C 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.

%C This sequence is not increasing: a(10) = 2450 < a(9) = 3600.

%C If gcd(a, b) = 1, then number of squares z = a*b * (a^2+b^2) is in A344334.

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

%C Every term is even.

%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 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.

%e       ___ ___ _ ___ ___ _

%e      |   |   |_|   |   |_|

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

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

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

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

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

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

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

%e      |   |   |_|   |   |_|

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

%e 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.

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

%K nonn

%O 1,1

%A _Bernard Schott_, Jun 13 2021