A345287 -id:A345287 - cp's OEIS frontend

A345286 a(n) is the number of large or small squares that are used to tile primary squares of type 1 (see A344331) whose side length is A345285(n).

20, 90, 272, 468, 650, 1280, 1332, 2900, 3600, 2450, 7650, 5760, 4160, 6642, 10388, 810, 16400, 10100, 1088, 25578, 29952, 14762, 27540, 20880, 42048, 50960, 54900, 41600, 28730, 65610, 81920, 90650, 60500, 38612, 98100, 50850, 125712, 85248, 142400, 149940

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 z = (a*b)^2 * (a^2+b^2) = a*b*s with a < b.
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.
This sequence is not increasing: a(10) = 2450 < a(9) = 3600.
If gcd(a, b) = 1, then number of squares z = a*b * (a^2+b^2) is in A344334.
If a = 1 and b = n > 1, then number of squares z = n^2 * (n^2+1) is in A071253 \ {0,2}.
Every term is even.

			The primary square with side A345285(1) = 10 can be tiled with a(1) = 20 small squares of side a = 1 and 20 large squares of side b = 2.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
The primary square with side A345285(6) = 160 can be tiled with a(6) = 1280 small squares of side a = 2 and 1280 large squares of 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. A071253, A344330, A344331, A344333, A344334, A345285, A345287.

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.

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

A345286 a(n) is the number of large or small squares that are used to tile primary squares of type 1 (see A344331) whose side length is A345285(n).

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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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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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