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).

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

Author

Bernard Schott, Jul 13 2021

Keywords

• nonn
• tabf

Comments

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).

Examples

			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:
     ________ ________ ________ _____
    |        |        |        |     |
    |        |        |        |     |
    |        |        |        |_____|
    |________|________|________|     |
    |        |        |        |     |
    |        |        |        |_____|
    |        |        |        |     |
    |________|________|________|     |
    |        |        |        |_____|
    |        |        |        |     |
    |        |        |        |     |
    |_____ __|___ ____|_ ______|_____|
    |     |      |      |      |     |
    |     |      |      |      |     |
    |_____|______|______|______|_____|
		

Crossrefs

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

Cf. A345286 (similar for type 1).