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).

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

Author

Bernard Schott, Aug 11 2021

Keywords

• nonn

Comments

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.

Examples

			-> 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.
		

References

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

Links

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

Crossrefs

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

Programs

PARI

\\ 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

Formula

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).

Extensions

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