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.

10, 30, 68, 78, 130, 222, 290, 300, 350, 510, 520, 738, 742, 820, 1010, 1218, 1342, 1530, 1740, 1752, 1820, 1830, 2210, 2590, 2750, 2758, 3270, 3390, 3492, 3560, 3570, 4112, 4290, 4498, 4770, 4930, 5850, 6028, 6328, 6870, 6878, 6942, 8020, 8030, 8190, 8610, 9282, 9620, 9962

Offset

1,1

Comments

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.

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).

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.

If a = 1 and b = n > 1, then sides of squares s = n * (n^2+1) form the subsequence A034262 \ {0, 1}.

Every term is even.

References

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

Links

Table of n, a(n) for n=1..49.

Index to sequences related to Olympiads.

Example

Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|

Prog

(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

Crossrefs

Cf. A034262, A344330, A344331, A344332, A344334.

Sequence in context: A104044 A124080 A345285 * A034127 A229466 A269261

Adjacent sequences: A344330 A344331 A344332 * A344334 A344335 A344336

Keyword

nonn

Author

Bernard Schott, Jun 01 2021

Status

approved