cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A344330 Sides s of squares that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

10, 15, 20, 30, 40, 45, 50, 60, 65, 68, 70, 75, 78, 80, 90, 100, 105, 110, 120, 130, 135, 136, 140, 150, 156, 160, 165, 170, 175, 180, 190, 195, 200, 204, 210, 220, 222, 225, 230, 234, 240, 250, 255, 260, 270, 272, 280, 285, 290, 300, 310, 312, 315, 320, 325, 330, 340, 345, 350, 360, 369, 370
Offset: 1

Views

Author

Bernard Schott, May 15 2021

Keywords

Comments

This sequence is a generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008.
Some notations: 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.
Side s of such tiled squares must satisfy the Diophantine equation s^2 = z * (a^2+b^2).
There are two types of solutions. See A344331 for type 1 and A344332 for type 2.
If q is a term, k * q is another term for k > 1.

Examples

			-> Example of type 1:
Square 10 x 10 with a = 1, b = 2, s = 10, z = 20.
      ___ ___ _ ___ ___ _
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_| with 10 elementary 2 x 5 rectangles
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|              ___ ___ _
     |___|___|_|___|___|_|             |   |   |_|
     |   |   |_|   |   |_|             |___|___|_|
     |___|___|_|___|___|_|
     |   |   |_|   |   |_|
     |___|___|_|___|___|_|
.
-> Example of type 2:
Square 15 x 15 with a = 3, b = 4, s = 15, z = 9.
      ________ ________ ________ _____
     |        |        |        |     |
     |        |        |        |     |
     |        |        |        |_____|
     |_______ |________|________|     |
     |        |        |        |     |
     |        |        |        |_____|
     |        |        |        |     |
     |________|________|________|     |
     |        |        |        |_____|
     |        |        |        |     |
     |        |        |        |     |
     |_____ __|___ ____|_ ______|_____|
     |     |      |      |      |     |
     |     |      |      |      |     |
     |_____|______|______|______|_____|
Remarks:
- With terms as 10, 20, ... we only obtain sides of squares of type 1:
10 is a term of this type because the square 10 X 10 only can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2 (see first example),
20 is another term of this type because the square 20 X 20 only can be tiled with 80 squares of size 1 x 1 and 80 squares of size 2 x 2.
- With terms as 15, 65, ... we only obtain sides of squares of type 2:
15 is a term of this type because the square 15 X 15 only can be tiled with 9 squares of size 3 X 3 and 9 squares of size 4 X 4 (see second example),
65 is another term of this type because the square 65 X 65 only can be tiled with 25 squares of size 5 X 5 and 25 squares of size 12 X 12.
- With terms as 30, 60, ... we obtain both sides of squares of type 1 and of type 2:
30 is a term of type 1 because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2, but,
30 is also a term of type 2 because the square 30 X 30 can be tiled with 9 squares of size 6 X 6 and 9 squares of size 8 X 8.

References

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

Links

Crossrefs

Subsequences: A008592 \ {0}, A008597 \ {0}, A034262 \ {0,1}.
Cf. A344331, A344332, A344333, A344334.

Programs

  • PARI
    pts(lim) = my(v=List(), m2, s2, h2, h); for(middle=4, lim-1, m2=middle^2; for(small=1, middle, s2=small^2; if(issquare(h2=m2+s2, &h), if(h>lim, break); listput(v, [small, middle, h])))); vecsort(Vec(v)); \\ A009000
isdp4(s) = my(k=1, x); while(((x=k^4 - (k-1)^4) <= s), if (x == s, return (1)); k++); return(0);
isokp2(s) = {if (!isdp4(s), return(0)); if (s % 2, my(vp = pts(s)); for (i=1, #vp, my(vpi = vp[i], a = vpi[1], b = vpi[2], c = vpi[3]); if (a*c/(c-b) == s, return(1)); ); ); }
isok2(s) = {if (isokp2(s), return (1)); fordiv(s, d, if ((d>1) || (dx*y*(x^2+y^2), [1..m]), s);}
isok1(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Jun 04 2021

Extensions

Corrected by Michel Marcus, May 18 2021
Incorrect term 145 removed by Michel Marcus, Jun 04 2021

A344331 Side s of squares of type 1 that can be tiled with squares of two different sizes so that the number of large or small squares is the same.

Original entry on oeis.org

10, 20, 30, 40, 50, 60, 68, 70, 78, 80, 90, 100, 110, 120, 130, 136, 140, 150, 156, 160, 170, 180, 190, 200, 204, 210, 220, 222, 230, 234, 240, 250, 260, 270, 272, 280, 290, 300, 310, 312, 320, 330, 340, 350, 360, 370, 380, 390, 400, 408, 410, 420, 430, 440, 444, 450, 460, 468, 470
Offset: 1

Views

Author

Bernard Schott, May 20 2021

Keywords

Comments

This sequence is relative to the generalization of the 4th problem proposed for the pupils in grade 6 during the 19th Mathematical Festival at Moscow in 2008 (see A344330).
There are two types of solutions, the first one is proposed here, while type 2 is described in A344332.
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.
-> Primitive squares
Side s of primitive squares of type 1 must satisfy the Diophantine equation s^2 = z * (a^2+b^2), with gcd(a, b) = 1, and without using the conditions a^2+b^2 = c^2, when a and b belong to a Pythagorean triple (a, b, c).
In this case, the sides of the primitive squares of type 1 are s = a*b * (a^2+b^2) with 1 <= a < b and gcd(a, b) = 1 (A344333), then corresponding z = (a*b)^2 * (a^2+b^2) (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.
In particular: for a = 1, b = n, s = n*(n^2+1) form the subsequence A034262 \ {0, 1} and z = n^2*(n^2+1) form the subsequence A071253 \ {0, 2}).
See example with design for a square of side s = 10 with a = 1, b = 2, m = 10, z = 20.
-> Non-primitive squares
If s is the side of a primitive square of type 1 with z squares of side a and z squares of side b, then every k * s is a non-primitive term that gives one or two distinct tilings of type 1, depending of value of k:
- For every k > 1, the square ks X ks can be tiled with k^2*z squares of side a and k^2*z squares of side b (see example).
- For every k = r^4, r>1, the square ks X ks also can be tiled with z squares of side ka and z squares of side kb.
---> Consequences:
1) For every pair (a, b), 1 <= a < b, there is a square of side s = a*b * (a^2+b^2) that can be tiled with squares of side a and side b so that the number z of squares of side a and side b is the same, this number z = (a*b)^2 * (a^2+b^2).
2) If q is a term and K > 1, K * q is another term.
3) Every term is even.

Examples

			Primitive square with s = 10:
   a = 1, b = 2, s = 10, m = 10, z = 20, and
Non-primitive square with s = 20:
   a = 1, b = 2, s = 20, m = 40, z = 80.
      ___ ___ _ ___ ___ _ ___________________
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|                   |
     |   |   |_|   |   |_|                   |
     |___|___|_|___|___|_|___________________|
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |                   |                   |
     |___________________|___________________|
with respectively m = 10 (and m = 40) elementary 2 X 5 rectangles as below:
          ___ ___ _
         |   |   |_|
         |___|___|_|
There are these three possibilities:
- 10 is a primitive term because the square 10 X 10 can be tiled with 20 squares of size 1 X 1 and 20 squares of size 2 X 2, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 2 X 2.
- 20 is a non-primitive term because the square 20 X 20 can be tiled with 80 squares of size 1 X 1 and 80 squares of size 2 X 2.
- 30 is a primitive term because the square 30 X 30 can be tiled with 90 squares of size 1 X 1 and 90 squares of size 3 X 3, and no smaller square can be tiled with a same number of squares of size 1 X 1 and of squares of size 3 X 3,
  but also, 30 is a non-primitive term because the square 30 X 30 can be tiled with 180 squares of size 1 X 1 and 180 squares of size 2 X 2.

References

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

Links

Crossrefs

Cf. A344330, A344332, A344333, A344334.
Cf. A034262, A071253.

Programs

  • PARI
    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); }
isok(s) = {if (isokp1(s), return (1)); fordiv(s, d, if ((d>1) || (dMichel Marcus, Dec 22 2021

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.

Original entry on oeis.org

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

Views

Author

Bernard Schott, Jun 01 2021

Keywords

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.

Examples

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

References

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

Links

Crossrefs

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

Programs

  • 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

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).

Original entry on oeis.org

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

Views

Author

Bernard Schott, Jun 13 2021

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

Cf. A071253, A344330, A344331, A344333, A344334, A345285, A345287.

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.

Original entry on oeis.org

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

Views

Author

Bernard Schott, Jun 14 2021

Keywords

Comments

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).

Examples

			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).

References

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

Links

Crossrefs

Cf. A344330, A344331, A344333, A344334, A345285, A345286, A346264.

Programs

  • PARI
    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

Formula

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

Extensions

Corrected and extended by Michel Marcus, Dec 22 2021

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

Views

Author

Bernard Schott, Jun 13 2021

Keywords

Comments

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.

Examples

			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.

References

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

Links

Crossrefs

Cf. A034262, A344330, A344333, A344334, A345286, A345287.
Subsequence of A344331.
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.