cp's OEIS Frontend

A344333 Primitive side of squares of type 1 (A344331) that are tiled with squares of two different sizes so that the number of large or small squares is the same.

%I A344333 #16 Dec 23 2021 10:03:42
%S A344333 10,30,68,78,130,222,290,300,350,510,520,738,742,820,1010,1218,1342,
%T A344333 1530,1740,1752,1820,1830,2210,2590,2750,2758,3270,3390,3492,3560,
%U A344333 3570,4112,4290,4498,4770,4930,5850,6028,6328,6870,6878,6942,8020,8030,8190,8610,9282,9620,9962
%N A344333 Primitive side of squares of type 1 (A344331) that are tiled with squares of two different sizes so that the number of large or small squares is the same.
%C A344333 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 A344333 Every term is of the form s = a*b * (a^2+b^2) with gcd(a, b) = 1, then corresponding z = (a*b)^2 * (a^2+b^2) (see A344334).
%C A344333 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 A344333 If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.
%C A344333 Every term is even.
%D A344333 Ivan Yashchenko, Invitation to a Mathematical Festival, pp. 10 and 102, MSRI, Mathematical Circles Library, 2013.
%H A344333 <a href="/index/O#Olympiads">Index to sequences related to Olympiads</a>.
%e A344333 Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
%e A344333       ___ ___ _ ___ ___ _
%e A344333      |   |   |_|   |   |_|
%e A344333      |___|___|_|___|___|_|
%e A344333      |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
%e A344333      |___|___|_|___|___|_|
%e A344333      |   |   |_|   |   |_|              ___ ___ _
%e A344333      |___|___|_|___|___|_|             |   |   |_|
%e A344333      |   |   |_|   |   |_|             |___|___|_|
%e A344333      |___|___|_|___|___|_|
%e A344333      |   |   |_|   |   |_|
%e A344333      |___|___|_|___|___|_|
%o A344333 (PARI) isok(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->if (gcd(x,y)==1, x*y*(x^2+y^2), 0), [1..m]), s); } \\ _Michel Marcus_, Dec 22 2021
%Y A344333 Cf. A034262, A344330, A344331, A344332, A344334.
%K A344333 nonn
%O A344333 1,1
%A A344333 _Bernard Schott_, Jun 01 2021