A346263 -id:A346263 - cp's OEIS frontend

A344332 Side s of squares of type 2 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

15, 30, 45, 60, 65, 75, 90, 105, 120, 130, 135, 150, 165, 175, 180, 195, 210, 225, 240, 255, 260, 270, 285, 300, 315, 325, 330, 345, 350, 360, 369, 375, 390, 405, 420, 435, 450, 455, 465, 480, 495, 510, 520, 525, 540, 555, 570, 585, 600, 615, 630, 645, 650, 660, 671, 675, 690, 700, 705, 715, 720, 735, 738, 750, 765, 780, 795, 810, 825, 840, 845, 855, 870, 875, 885, 900

Offset: 1

Bernard Schott, May 20 2021

nonn

This sequence is relative to the generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008 (see A344330).
There are two types of solutions, the second one is proposed here, while type 1 is described in A344331.
If m is a term and k > 1, k * m is another term.
Every term (primitive or not primitive) is the side of an elementary square of type 2 (see A346263).
Some notations: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
-> Primitive squares
Side s of primitive squares of type 2 must satisfy the Diophantine equation s^2 = z * (a^2+b^2) with the conditions a^2+b^2 = c^2 and gcd(a, b, c) = 1.
In this case, q = a/(c-b) must be odd, and side s = q*c = a*c/(c-b) = (a+b)*c/a with a number of squares z = q^2 = (a/(c-b))^2 = ((b+c)/a)^2.
Indeed, these conditions give exactly the following solutions for n >= 2: s = n^4-(n-1)^4 (A005917), a = 2*n-1 (A005408), b = 2*n*(n-1) (A046092), c = 2*n*(n-1)+1 (A001844), z = (2*n-1)^2 (A016754); this results come from the identity:
   (n^4 - (n-1)^4)^2 = (2*n-1)^2 * ((2*n-1)^2 + (2*n*(n-1))^2).
For n >= 2, every primitive square is composed by a square S1 of z = (2*n-1)^2 large squares with side b = 2*n*(n-1), then an edge on two sides of this square S1 of z = (2*n-1)^2 small squares with side a = 2*n-1.
See example with design of square of side s = 15 with a = 3, b = 4, c = 5, q = 3, z = 9, obtained with n= 2.
-> Non-primitive squares
If s is the side of a primitive square of type 2, then every k * s, k > 1 is a non-primitive term that gives two distinct tilings of type 2.
The square ks X ks can be tiled with z = q^2 = (2n-1)^2 = (a/(c-b))^2 = ((b+c)/a)^2 squares of side ka and of side kb, but also,
The square ks X ks can be tiled with z = k^2*q^2 = ((2n-1)*k)^2 = (k*a/(c-b))^2 = (k*(b+c)/a)^2 squares of side a and of side b (see example).

			Primitive square with s = 15:
  a = 3, b = 4, c = 5, s = 15, z = 9; s = 15 is the side of primitive square, with  z = 9 squares of size 3 x 3 and 9 squares of size 4 x 4
Non-primitive square k*s = 2*15 = 30:
  a = 3, b = 4, c = 5, s = 30, z = 36, this square is obtained with 4 copies of the primitive square as below.
  a = 6, b = 8, c = 10, s = 30, z = 9, this square and its tiling are exactly as the primitive square with scale 2.
               b = 4 (or = 8)     a = 3 (or = 6)
            ________ ________ ________ ______ ______________________________
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           |        |        |        |______|                              |
           |_______ |________|________|      |                              |
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           |        |        |        |______|                              |
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           |________|________|________|      |                              |
           |        |        |        |______|                              |
           |        |        |        |      |                              |
           |        |        |        |      |                              |
           |_____ __|___ ____|_ ______|______|                              |
           |     |      |      |      |      |                              |
           |     |      |      |      |      |                              |
           |_____|______|______|______|______|______________________________|
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           |_________________________________|______________________________|
                      s = 15               s = 30

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

Index to sequences related to Olympiads.

Cf. A001844, A005408, A005917, A046092, A046754, A344330, A344331.

pts(lim) = my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [small, middle, h])))); vecsort(Vec(v)); \\ A009000
isdp4(s) = my(k=1, x); while(((x=k^4 - (k-1)^4) <= s), if (x == s, return (1)); k++); return(0);
isokp2(s) = {if (!isdp4(s), return(0)); if (s % 2, my(vp = pts(s)); for (i=1, #vp, my(vpi = vp[i], a = vpi[1], b = vpi[2], c = vpi[3]); if (a*c/(c-b) == s, return(1)); ); ); }
isok2(s) = {if (isokp2(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Jun 04 2021

A346264 a(n) is the number of distinct possible tilings of type 2 for squares with side = A344332(n) and that can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.

Original entry on oeis.org

1, 2, 2, 3, 1, 2, 4, 2, 4, 2, 3, 4, 2, 1, 6, 4, 4, 4, 5, 2, 3, 6, 2, 6, 4, 2, 4, 2, 2, 8, 1, 3, 8, 4, 6, 2, 8, 2, 2, 6, 4, 4, 4, 6, 9, 2, 4, 7, 8, 2, 8, 2, 4, 6, 1, 6, 4, 3, 2, 2, 10, 3, 2, 6, 4, 12, 2, 8, 4, 8, 2, 4, 4, 2, 2, 12, 4, 2, 4, 6, 7, 8, 8, 2, 6, 4, 5, 12, 2, 12, 2, 3, 3, 4

Offset: 1

Bernard Schott, Aug 09 2021

nonn

Every side of square of type 2 in A344332 is also the side of an elementary square of type 2. An elementary square of type 2 is the smallest square that can be tiled with squares of two different sides a < b satisfying a^2+b^2 = c^2 and so that the numbers of small and large squares are equal.
Some notation: s = side of the tiled square, a = side of small squares, b = side of large squares, and z = number of small squares = number of large squares.
a(n) = 1 iff A344332(n) is a term of A005917 that is not a multiple of another term of A005917 (15, 65, 175, 369, 671, 2465, ...).
The first side that is a multiple of two primitive sides is 195 = 13*15 = 3*65 (see 3rd example).

			---> For a(1), A344332(1) = 15, then, with the formula, we get a(1) = tau(A344332(1)/A005917(2)) = tau(15/15) = tau(1) = 1, and the corresponding tiling of this smallest square 15 X 15 of type 2 consists of z = 9 squares whose sides (a,b) = (3,4) (see below).
            ________ ________ ________ ______
           |        |        |        |      |
           |        |        |        |      |
           |        |        |        |______|
           |_______ |________|________|      |
           |        |        |        |      |
           |        |        |        |______|
           |        |        |        |      |
           |________|________|________|      |
           |        |        |        |______|
           |        |        |        |      |
           |        |        |        |      |
           |_____ __|___ ____|_ ______|______|
           |     |      |      |      |      |
           |     |      |      |      |      |
           |_____|______|______|______|______|
                       a(1) = 1
---> For a(2), A344332(2) = 30, then, with the formula, we get a(2) = tau(A344332(2)/A005917(2)) = tau(30/15) = tau(2) = 2, and these 2 distinct tilings are:
1) 30 = 2*A344332(1) = 2*15, z(30) = 2^2 * z(15) = 4*9 = 36 and square 30 X 30 can be tiled with z = 36 squares whose sides (a,b) = (3,4), that is 4 copies of the elementary and primitive square 15 X 15 (as above). Also,
2) 30 = 1*A344332(2) = 1*30, z(30) = 1^2 * z(15) = 1*9 = 9 and the elementary square 30 X 30 can be tiled with z = 9 squares whose sides (a,b) = (6,8) (see link with corresponding drawings).
---> For a(16), A344332(16) = 195, then, with the formula, we get a(16) = tau(A344333(16)/A005917(2)) + tau(A344333(16)/A005917(3)) = tau(195/15) + tau(195/65) = tau(13) + tau(3) = 2+2 = 4, and these 4 distinct tilings are:
1) 195 = 13*A344332(1) = 13*15, z_1(195) = 13^2 * z(15) = 169*9 = 1521 and square 195 X 195 can be tiled with z = 1521 squares whose sides (a,b) = (3,4), that is 169 copies of the elementary and primitive square 15 X 15, as above;
2) 195 = 1*A344332(16) = 1*195, z_2(195) = 1^2 * z(195) = 1*9 = 9 and the elementary square 195 X 195 can be tiled with z = 9 squares whose sides (a,b) = (39,52);
3) 195 = 3*A344332(5) = 3*65, z_3(195) = 3^2 * z(65) = 9*25 = 225 [z(65) = A346263(5) = T(5,1) = 25] and square 195 X 195 can be tiled with z = 225 squares whose sides (a,b) = (5,12), that is 9 copies of the elementary and primitive square 65 X 65;
4) 195 = 1*A344332(16) = 1*195, z_4(195) = 1^2 * z(195) = 1*25 = 25 and the elementary square 195 X 195 can be tiled with z = 25 squares whose sides (a,b) = (15,36).

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

Bernard Schott, Tilings of type 2 for square 30 X 30 with a(2) = 2.

Cf. A005917, A016754, A344332, A346263.

\\ using isok2 from A344332; isok7 is for A005917
isok7(k) = my(kk= sqrtnint(k\4, 3)+2); vecsearch(vector(kk, i, (i+1)^4 - i^4), k);
sd(x) = sumdiv(x, d, if (isok7(d), numdiv(x/d)));
lista(nn) = my(v2 = select(isok2, [1..nn])); apply(sd, v2);

a(n) = Sum_{(k>=2) & (A005917(k)|A344332(n))} tau(A344332(n)/A005917(k)).

cp's OEIS Frontend

A344332 Side s of squares of type 2 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

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