A065072 -id:A065072 - cp's OEIS frontend

A360773 Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.

0, 1, 8, 1024, 620448

Offset: 1

Scott R. Shannon and N. J. A. Sloane, Feb 20 2023

All possible tilings are counted, including those identical by symmetry. Note that distinct dimensions means that, for example, a 1 x 3 rectangle can only be used once, regardless of if it lies horizontally or vertically.

Only squares with even edges lengths are possible since the area of a square with odd edge lengths is odd, while the perimeter of any rectangle is even.

			a(1) = 0 as a 2 x 2 square, with area 4, cannot be tiled with distinct rectangles with perimeters that sum to 4.
a(2) = 1 as a 4 x 4 rectangle, with area 16, can be tiled with a 4 x 4 square with perimeter 4 + 4 + 4 + 4 = 16.
a(3) = 8. The possible tilings for the 6 x 6 square, with area 36, excluding those equivalent by symmetry, are:
.
  +---+---+---+---+---+---+   +---+---+---+---+---+---+
  |                       |   |                       |
  +---+---+---+---+---+---+   +                       +
  |                       |   |                       |
  +                       +   +---+---+---+---+---+---+
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +                       +   +                       +
  |                       |   |                       |
  +---+---+---+---+---+---+   +---+---+---+---+---+---+
.
where for the first tiling (2*6 + 2*1) + (2*6 + 2*5) = 36 while for the second tiling (2*6 + 2*2) + (2*6 + 2*4) = 36. Both of these tilings can occur in 4 ways, giving 8 ways in total.
a(4) = 1024. And example tiling of the 8 x 8 square, with area 64, is:
.
  +---+---+---+---+---+---+---+---+
  |   |                   |       |
  +   +                   +---+---+
  |   |                   |       |
  +   +                   +       +
  |   |                   |       |
  +---+---+---+---+---+---+---+---+
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +                               +
  |                               |
  +---+---+---+---+---+---+---+---+
.
where (2*1 + 2*3) + (2*5 + 2*3) + (2*2 + 2*1) + (2*2 + 2*2) + (2*8 + 2*5) = 64.

Cf. A360499, A360498, A360725, A360256, A182275, A004003, A099390, A065072.

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pij/n)^2 + sin(Pik/n)^2).

Original entry on oeis.org

0, 1, 96, 93789, 1244160000, 241885578271872, 700566272328037500000, 30323548995402141685610526683, 19627362048402730985830806120284160000, 189995156103157091521654945902925881881155376920, 27506190205802587152768139358989866456457087869970721213256

Offset: 0

Vaclav Kotesovec, Jan 06 2021

nonn

Cf. A004003, A007726, A065072, A067518, A127605, A340052, A340165, A340182, A340183, A340293.

Table[2^(n^2 - 1) * Product[1 + Sin[Pi*j/n]^2 + Sin[Pi*k/n]^2, {j, 1, n}, {k, 1, n}], {n, 0, 10}] // Round

a(n) = 2^(n^2-1) * Product_{j=1..n, k=1..n} (3 - cos(Pi*j/n)^2 - cos(Pi*k/n)^2).
a(n) = 2^(n^2-1) * Product_{j=1..n, k=1..n} (2-cos(2*Pi*j/n)/2-cos(2*Pi*k/n)/2).
a(n) ~ 2^(n^2-1) * exp(4*c*n^2/Pi^2), where c = Integral_{x=0..Pi/2, y=0..Pi/2} log(1 + sin(x)^2 + sin(y)^2) dy dx = -Pi^2*(log(2) + log(sqrt(2)-1)/2) + Pi * Integral_{x=0..Pi/2} log(1 + sqrt(1 + 1/(1 + sin(x)^2))) dx = A340421 = 1.627008991085721315763766677017604437985734719035793082916212355323520649...

A143234 a(n) = sqrt(2^(-n)*A004003(n)) mod 32.

Original entry on oeis.org

1, 1, 3, 29, 5, 5, 7, 25, 9, 9, 11, 21, 13, 13, 15, 17, 17, 17, 19, 13, 21, 21, 23, 9, 25, 25, 27, 5, 29, 29, 31, 1, 1, 1, 3, 29, 5, 5, 7, 25, 9, 9, 11, 21, 13, 13, 15, 17, 17, 17, 19, 13, 21, 21, 23, 9, 25, 25, 27, 5, 29, 29, 31, 1, 1, 1, 3, 29, 5, 5, 7, 25, 9, 9, 11, 21, 13, 13, 15, 17

Offset: 0

Eric W. Weisstein, Jul 31 2008

G. C. Greubel, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Domino Tiling
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A004003, A065072, A109613, A121262.

A143234:= func< n | (-1)^(0^((n+1) mod 4))*(2*Floor(n/2) + 1) mod 32 >;
[A143234(n): n in [0..100]]; // G. C. Greubel, Sep 11 2024

(* First program *)
a[n_]:= Mod[If[EvenQ[n], n + 1, (-1)^((n-1)/2)*n], 32];
Table[a[n], {n,0,100}]
(* Second program *)
A143234[n_]:= Mod[(-1)^(Floor[Mod[n,4]/3])*(2*Floor[n/2]+1), 32];
Table[A143234[n], {n,0,100}] (* G. C. Greubel, Sep 12 2024 *)

def A143234(n): return ((-1)^(0^((n+1)%4))*(2*int(n//2)+1))%32
[A143234(n) for n in range(101)] # G. C. Greubel, Sep 11 2024

a(n) = A065072(n) mod 32.
From G. C. Greubel, Sep 12 2024: (Start)
a(n) = ( (-1)^A121262(n+1) * A109613(n) ) mod 32.
a(n) = a(n-32). (End)

Offset changed by Editor(s) of Oeis.

A340168 Decimal expansion of a constant related to the asymptotics of A004003.

Original entry on oeis.org

1, 1, 0, 8, 8, 6, 2, 2, 5, 8, 7, 8, 0, 7, 6, 7, 5, 1, 3, 2, 7, 6, 9, 5, 1, 1, 6, 2, 1, 3, 0, 8, 1, 9, 2, 9, 2, 6, 4, 5, 2, 6, 6, 1, 2, 6, 9, 6, 3, 5, 6, 9, 2, 2, 4, 3, 6, 2, 9, 4, 3, 1, 4, 1, 8, 4, 4, 7, 3, 5, 5, 6, 5, 3, 0, 9, 3, 4, 8, 6, 6, 3, 2, 1, 3, 4, 3, 9, 7, 1, 4, 6, 7, 5, 0, 7, 9, 0, 1, 5, 5, 7, 4, 0, 5

Offset: 1

Vaclav Kotesovec, Dec 30 2020

			1.1088622587807675132769511621308192926452661269635692243629431418447355653...

Cf. A004003, A007341, A007726, A065072, A127605, A340052.

RealDigits[2*E^(Catalan/Pi)/(1 + Sqrt[2]), 10, 110][[1]]

Equals lim_{n->infinity} A004003(n) / ((sqrt(2)-1)^(2*n) * exp(4*G*n*(n+1)/Pi)), where G is the Catalan's constant A006752.

Equals 2*exp(G/Pi) / (1 + sqrt(2)), where G is Catalan's constant A006752.

cp's OEIS Frontend

A360773 Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.

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A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pij/n)^2 + sin(Pik/n)^2).

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A143234 a(n) = sqrt(2^(-n)*A004003(n)) mod 32.

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A340168 Decimal expansion of a constant related to the asymptotics of A004003.

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cp's OEIS Frontend

A360773 Number of ways to tile a 2n X 2n square using rectangles with distinct dimensions such that the sum of the rectangles perimeters equals the area of the square.

Views

Author

Keywords

Comments

Examples

Crossrefs

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pi*j/n)^2 + sin(Pi*k/n)^2).

Views

Author

Keywords

Crossrefs

Programs

Mathematica

Formula

A143234 a(n) = sqrt(2^(-n)*A004003(n)) mod 32.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

SageMath

Formula

Extensions

A340168 Decimal expansion of a constant related to the asymptotics of A004003.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A340396 a(n) = 2^(n^2 - 1) * Product_{j=1..n, k=1..n} (1 + sin(Pij/n)^2 + sin(Pik/n)^2).