A345286 -id:A345286 - cp's OEIS frontend

A345287 a(n) is the number of distinct possible tilings of type 1 for squares with side = A344331(n) and that can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.

1, 2, 3, 3, 2, 6, 1, 2, 1, 4, 5, 4, 2, 9, 3, 2, 4, 6, 2, 5, 2, 10, 2, 6, 2, 6, 4, 1, 2, 2, 12, 3, 6, 7, 3, 6, 3, 13, 2, 3, 6, 6, 6, 5, 15, 2, 4, 10, 8, 4, 2, 12, 2, 6, 2, 10, 4, 4, 2, 2, 15, 3, 6, 7, 10, 2, 14, 4, 2, 4, 8, 6, 6, 2, 20, 2, 3, 4, 4, 10, 7, 6, 12, 2, 2, 10, 6

Offset: 1

Bernard Schott, Jun 14 2021

nonn

Every side of square of type 1 in A344331 is also the side of an elementary square of type 1. An elementary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b 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 A344331(n) is a term of A344333 that is not a multiple of another term of A344333 (10, 68, 78, 222, ...).
The first side that is a multiple of two primitive sides is 30 = 3*10 = 1*30 (see 2nd example).

			For a(1), A344331(1) = 10, then, with the formula, we get a(1) = tau(A344331(1)/A344333(1)) = tau(10/10) = tau(1) = 1. This unique corresponding tiling of this square 10 x 10 of type 1 with side s = 10 consists of z = 20 squares whose sides (a,b) = (1,2) (see below).
          ___ ___ _ ___ ___ _
         |   |   |_|   |   |_|
         |___|___|_|___|___|_|
         |   |   |_|   |   |_|
         |___|___|_|___|___|_|
         |   |   |_|   |   |_|
         |___|___|_|___|___|_|
         |   |   |_|   |   |_|
         |___|___|_|___|___|_|
         |   |   |_|   |   |_|
         |___|___|_|___|___|_|
                a(1) = 1
For a(3), A344331(3) = 30, then, with the formula, we get a(3) = tau(A344331(3)/A344333(1)) + tau(A344331(3)/A344333(2)) = tau(30/10) + tau(30/30) = tau(3) + tau(1) = 3. The 3 corresponding tilings of the square 30 x 30 of type 1 with side s = 30 consists of:
-> from 30 = 3*A344333(1) = 3*10, square with side s = 30 can be tiled with z = 180 squares with sides (a,b) = (1,2), indeed with 9 copies of primitive square 10 x 10, as above.
-> from 30 = 1*A344331(3) = 1*30, square with side s = 30 can be tiled with z = 20 squares with sides (a,b) = (3,6), indeed, it is the above square with scale 3.
-> from 30 = 1*A344331(3) = 1*30, square with side s = 30 can also be tiled with z = 90 squares with sides (a,b) = (1,3), indeed that is primitive square 30 x 30 with squares (a,b) = (1,3).

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

Index to sequences related to Olympiads.

Cf. A344330, A344331, A344333, A344334, A345285, A345286, A346264.

isokp1(s) = {if (!(s % 2) && ispower(s/2, 4), return (0)); my(m = sqrtnint(s, 3)); vecsearch(setbinop((x, y)->x*y*(x^2+y^2), [1..m]), s); }
isok1(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dA344331
isok3(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); } \\ A344333
sd(x) = sumdiv(x, d, if (isok3(d), numdiv(x/d)));
lista(nn) = my(v1 = select(isok1, [1..nn])); apply(sd, v1); \\ Michel Marcus, Dec 22 2021

a(n) = Sum_{(k=1..n) & (A344333(k)|A344331(n))} tau(A344331(n)/A344333(k)).

a(n) = Sum_{(d | A344331(n)) & (d in A344333)} tau(A344331(n)/d) where tau is A000005. - Michel Marcus, Dec 22 2021

Corrected and extended by Michel Marcus, Dec 22 2021

A345285 Sides of primary squares of type 1 (A344331). A primary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b, so that the numbers of small and large squares are equal.

Original entry on oeis.org

10, 30, 68, 78, 130, 160, 222, 290, 300, 350, 480, 510, 520, 738, 742, 810, 820, 1010, 1088, 1218, 1248, 1342, 1530, 1740, 1752, 1820, 1830, 2080, 2210, 2430, 2560, 2590, 2750, 2758, 3270, 3390, 3492, 3552, 3560, 3570, 4112, 4290, 4498, 4640, 4770, 4800, 4930, 5508, 5600, 5850, 6028, 6250

Offset: 1

Bernard Schott, Jun 13 2021

nonn

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.
Every term is of the form s = a*b * (a^2+b^2), with 1 <= a < b, and corresponding z = (a*b)^2 * (a^2+b^2) (A345286).
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.
If gcd(a, b) = 1, then primitive sides of square s = a*b * (a^2+b^2) are in A344333 that is a subsequence.
If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.
If q is a term and integer r > 1, then q * r^4 is another term.
Every term is even.

			a(1) = 10 and the primary square 10 X 10 can be tiled with A345286(1) = 20 small squares with side a = 1 and 20 large squares with side b = 2.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|     with 10 elementary 2 X 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
a(6) = 160 is the first side of an primary square that is not primitive and it corresponds to (a,b) = (2,4); the square 160 X 160 can be tiled with A345286(6) = 1280 small squares with side a = 2 and 1280 large squares with side b = 4.

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

Index to sequences related to Olympiads.

Cf. A034262, A344330, A344333, A344334, A345286, A345287.

Subsequence of A344331.

A346263 Irregular triangle of numbers T(n,k) read by rows where each T(n,k) is the number of large or small squares that are used to tile elementary squares of type 2 whose length of side is A344332(n).

Original entry on oeis.org

9, 9, 9, 9, 25, 9, 9, 9, 9, 25, 9, 9, 9, 49, 9, 9, 25, 9, 9, 9, 9, 25, 9, 9, 9, 9, 25, 9, 9, 49, 9, 81, 9, 9, 25, 9, 9, 9, 9, 25, 9, 9, 9, 9, 25, 9, 49, 9, 9, 9, 9, 25, 9, 9, 9, 9, 25, 9, 121, 9, 9, 49, 9, 25, 9, 9, 81, 9, 9, 9, 25, 9, 9, 9, 9, 25, 9, 9, 49, 9, 9

Offset: 1

Bernard Schott, Jul 13 2021

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.
Every term is an odd square >= 9 and each odd square is present infinitely many times.
Notation: s_p (resp. s_e) = side of a primitive (resp. elementary) tiled square, a = side of small squares and b = side of large squares used to tile a primitive square, and z_p (z_e) = number of small squares = number of large squares used to tile a primitive (resp. elementary) square.
A primitive square with side s_p = a*c/(c-b) is tiled with z_p small and z_p large squares with sides a and b, and z_p = (a/(c-b))^2.
Each elementary square with a side s_e = k*s_p, k>0, is tiled with z_e small and z_e large squares with sides k*a and k*b, and z_e = z_p = (a/(c-b))^2.
When an elementary side A344332(n) is a multiple of m distinct primitive sides s_p, then there are m different values T(n,1), ..., T(n,m) in the row n (see example).

			The triangle T begins:
   n\k 1    2    3    4    5
   1:  9
   2:  9
   3:  9
   4:  9
   5: 25
   6:  9
   7:  9
   8:  9
   9:  9
  10: 25
  11:  9
  12:  9
  13:  9
  14: 49
  15:  9
  16:  9,   25
  17:  9
  ...
The first elementary side that is a multiple of two primitive sides (15 and 65) is A344332(16) = 195 = 13*15 = 3*65.
As 195 = 13*15, the number z of squares with side a = 13*3 = 39 and b = 13*4 = 52 to tile this elementary square is T(16,1) = (39/(65-52))^2 = 9.
As 195 = 3 * 65, the number z of squares with side a = 3*5 = 15 and b = 3*12 = 36 to tile this elementary square is T(16,2) = (15/(39-36))^2 = 25.
Hence, the elementary square with side A344332(16) = 195 has two different possible tilings: with T(16,1) = 9 squares of sides (a,b) = (39,52) or with T(16,2) = 25 squares of sides (a,b) = (15,36).
Elementary square 195 X 195 with a = 39, b = 52, s = 195, z = 9:
     ________ ________ ________ _____
    |        |        |        |     |
    |        |        |        |     |
    |        |        |        |_____|
    |________|________|________|     |
    |        |        |        |     |
    |        |        |        |_____|
    |        |        |        |     |
    |________|________|________|     |
    |        |        |        |_____|
    |        |        |        |     |
    |        |        |        |     |
    |_____ __|___ ____|_ ______|_____|
    |     |      |      |      |     |
    |     |      |      |      |     |
    |_____|______|______|______|_____|

Cf. A005917, A016754, A344330, A344332, A346264.

Cf. A345286 (similar for type 1).

cp's OEIS Frontend

A345287 a(n) is the number of distinct possible tilings of type 1 for squares with side = A344331(n) and that can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.

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A345285 Sides of primary squares of type 1 (A344331). A primary square of type 1 is the smallest square that can be tiled with squares of two different sides a < b, so that the numbers of small and large squares are equal.

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A346263 Irregular triangle of numbers T(n,k) read by rows where each T(n,k) is the number of large or small squares that are used to tile elementary squares of type 2 whose length of side is A344332(n).

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