A346264 -id:A346264 - 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

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).

A346265 a(n) is the number of distinct possible tilings of type 1 (A344331) or of type 2 (A344332) for a square whose side is A344330(n).

Original entry on oeis.org

1, 1, 2, 5, 3, 2, 2, 9, 1, 1, 2, 2, 1, 4, 9, 4, 2, 2, 13, 5, 3, 2, 4, 10, 2, 5, 2, 2, 1, 16, 2, 4, 6, 2, 10, 4, 1, 4, 2, 2, 17, 3, 2, 9, 13, 3, 6, 2, 3, 19, 2, 3, 4, 6, 2, 10, 6, 2, 7, 23, 2, 2, 3, 4, 18, 8, 4, 4, 2, 18, 2, 2, 6, 2, 18, 2, 4, 2, 4, 2, 2, 21, 3, 4, 6, 11, 14, 6, 2, 23

Offset: 1

Bernard Schott, Aug 11 2021

nonn

These squares with side = A344330(n) can be tiled with squares of two different sizes so that the numbers of large or small squares are equal.

Notation: 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.

			-> A344330(1) = A344331(1) = 10 and there is no k_2 such that A344330(1) = A344332(k_2) = 10, then a(1) = A345287(1) = 1 (example below of type 1):
   Primitive square 10 X 10 corresponding to a(1) = 1 with
    a = 1, b = 2, s = 10, z = 20:
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
-> A344330(2) = A344332(1) = 15 and there is no k_1 such that A344330(2) = A344331(k_1) = 15, then a(2) = A346264(1) = 1 (example below of type 2):
   Primitive square 15 X 15 corresponding to a(2) = 1 with
     a = 3, b = 4, c = 5, s = 15, z = 9:
        ________ ________ ________ ______
       |        |        |        |      |
       |        |        |        |      |
       |        |        |        |______|
       |_______ |________|________|      |
       |        |        |        |      |
       |        |        |        |______|
       |        |        |        |      |
       |________|________|________|      |
       |        |        |        |______|
       |        |        |        |      |
       |        |        |        |      |
       |_____ __|___ ____|_ ______|______|
       |     |      |      |      |      |
       |     |      |      |      |      |
       |_____|______|______|______|______|
-> A344330(4) = A344331(3) = A344332(2) = 30, then a(4) = A345287(3) + A346264(2) = 3+2 = 5 (see link with the corresponding 5 distinct tilings).
-> A344330(6) = A344332(3) = 45 and there is no k_1 such that A344330(6) = A344331(k_1) = 45, then a(6) = A346264(3) = 2.

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

Bernard Schott, The 5 distinct tilings corresponding to a(4) = 5.

Cf. A344330, A344331, A344332, A345287, A346264.

\\ isok1 from A344331 and isok2 from A344332
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
sd7(x) = sumdiv(x, d, if (isok3(d), numdiv(x/d))); \\ A345287
isok7(k) = my(kk= sqrtnint(k\4, 3)+2); vecsearch(vector(kk, i, (i+1)^4 - i^4), k); \\ A005917
sd4(x) = sumdiv(x, d, if (isok7(d), numdiv(x/d))); \\ A346264
lista(nn) = {for (n=1, nn, my(b1 = isok1(n), b2 = isok2(n)); if (b1 || b2, my(x = 0); if (b1, x += sd7(n)); if (b2, x += sd4(n)); print1(x, ", ");););} \\ Michel Marcus, Dec 23 2021

If A344330(n) = A344331(k_1) and there is no k_2 such that A344330(n) = A344332(k_2) then a(n) = A345287(k_1).
If A344330(n) = A344332(k_2) and there is no k_1 such that A344330(n) = A344331(k_1) then a(n) = A346264(k_2).
If A344330(n) = A344331(k_1) = A344332(k_2) then a(n) = A345287(k_1) + A346264(k_2).

a(19),a(59),a(86),a(87) corrected by Bernard Schott, Dec 23 2021

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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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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A346265 a(n) is the number of distinct possible tilings of type 1 (A344331) or of type 2 (A344332) for a square whose side is A344330(n).

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