A242889 -id:A242889 - cp's OEIS frontend

A242129 Number of tiles added in each step if a centrally symmetric Penrose rhomb tiling, beginning with a central 'Sun' configuration, is successively extended in steps as defined below.

5, 5, 10, 15, 20, 25, 40, 30, 30, 45, 60, 55, 65, 55, 70, 75, 90, 75, 90

Offset: 1

Felix Fröhlich, May 05 2014

In each step, add a new tile to each free edge of the current tiling. The first step is the placement of the five central tiles.

			See illustrations in Fröhlich, 2019.

David Austin, Penrose tiles talk across miles, AMS Feature Column, August 2005.
Felix Fröhlich, Layers in the Penrose rhomb tiling with a "Sun" patch as a center of global fivefold rotational symmetry, 2019.
Wikipedia, Penrose tiling

Cf. A242128 (5-fold, Star), A242888 (7-fold, Star), A242889 (7-fold, Sun), A242890 (8-fold, Star), A242891 (8-fold, Sun), A242892 (9-fold, Star), A242893 (9-fold, Sun), A242894 (Kite and dart, Star), A242895 (Kite and dart, Sun).

a(8)-a(19) from Felix Fröhlich, Jun 16 2019

A242128 Beginning with adding a central 'Star' configuration of a centrally symmetric Penrose rhomb tiling as the first step, number of tiles that can be added to the free edges of the current tiling.

Original entry on oeis.org

5, 10, 15, 15, 30, 25, 30, 50, 40, 40, 50, 60, 60, 75, 75

Offset: 1

Felix Fröhlich, May 05 2014

			See illustration in Fröhlich, 2019.

Felix Fröhlich, Layers in the Penrose rhomb tiling with a "Star" patch as a center of global fivefold rotational symmetry, 2019.
Wikipedia, Penrose tiling

Cf. A242129 (5-fold, Sun), A242888 (7-fold, Star), A242889 (7-fold, Sun), A242890 (8-fold, Star), A242891 (8-fold, Sun), A242892 (9-fold, Star), A242893 (9-fold, Sun), A242894 (Kite and dart, Star), A242895 (Kite and dart, Sun).

More terms from Felix Fröhlich, Jun 20 2019

A242888 Beginning with adding a central 'Star' configuration of a centrally symmetric rhomb tiling with 7-fold symmetry, number of tiles that can be added to the free edges of the tiling.

Original entry on oeis.org

7, 14, 21, 14, 14, 28

Offset: 1

Felix Fröhlich, May 25 2014

S. Weber, Fourier transform on quasiperiodic point sets (select '7-fold' with 'shift (n-1)/2n')

Cf. A242128, A242129, A242889.

A242894 Beginning with a central 'Star' configuration of a Penrose 'Kite' and 'Dart' tiling with rotational symmetry as the first step, number of tiles that can be added to the free edges of the current tiling.

Original entry on oeis.org

5, 10, 10, 20, 20, 25, 35, 40, 40, 45, 45

Offset: 1

Felix Fröhlich, May 25 2014

The initial 5 corresponds to the initial "star" made from 5 "darts". Given the 5-fold symmetry, all terms are divisible by 5. See the illustration for the first 8 terms. - M. F. Hasler, Jun 04 2019

M. F. Hasler, Illustration of first 8 terms of A242894, Jun 04 2019
Cole Leahy, Image of a symmetric patch of tiles with a 'Star' at the center (from a question about drawing subregions of Penrose tilings on TeX - LaTeX Stack Exchange), May 2013.

Cf. A242128, A242129, A242888, A242889, A242890, A242891, A242892, A242893, A242895.

A242895 Beginning with a central 'Sun' configuration of a Penrose 'Kite' and 'Dart' tiling with rotational symmetry as the first step, number of tiles that can be added to the free edges of the current tiling.

Original entry on oeis.org

5, 5, 10, 15, 20, 25, 30

Offset: 1

Felix Fröhlich, May 25 2014

The initial 5 corresponds to the "Sun" configuration made of 5 Penrose "Kites". In view of the 5-fold rotational symmetry, all terms are divisible by 5, see the illustrations given as LINKS. - M. F. Hasler, Jun 05 2019

Sam T. Benge(?), Image of 'Sun' and 'Star' configurations in a centrally symmetric 'Kite' and 'Dart' tiling, post to the POV-Ray Newsgroup, Oct 31 2009.
Wikipedia, Penrose tiling, Deflation for P2 and P3 tilings. (Defines "sun" and "star" configurations and generations corresponding to terms of this sequence.)

Cf. A242128, A242129, A242888, A242889, A242890, A242891, A242892, A242893, A242894.

A242890 Beginning with a central 'Star' configuration of a centrally symmetric arrangement of 8 rhombi with 8-fold rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Original entry on oeis.org

8, 16, 16, 16, 16, 16, 24

Offset: 1

Felix Fröhlich, May 25 2014

S. Weber, Fourier transform on quasiperiodic point sets (select '8-fold' with 'shift (n-1)/2n')

Cf. A242128, A242129, A242888, A242889, A242891, A242892, A242893, A242894, A242895.

A242891 Beginning with a centrally symmetric 'Sun' configuration of 8 rhombi with rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Original entry on oeis.org

8, 8, 8, 16, 24, 24, 40, 32, 48, 48, 40

Offset: 1

Felix Fröhlich, May 25 2014

S. Weber, Fourier transform on quasiperiodic point sets (select '8-fold' with 'shift 1/n')

Cf. A242128, A242129, A242888, A242889, A242890, A242892, A242893, A242894, A242895.

A242892 Beginning with a centrally symmetric 'Star' configuration of 9 rhombi with rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Original entry on oeis.org

9, 9, 18, 27, 27, 18

Offset: 1

Felix Fröhlich, May 25 2014

S. Weber, Fourier transform on quasiperiodic point sets (select '9-fold' with 'shift (n-1)/(2n)')

Cf. A242128, A242129, A242888, A242889, A242890, A242891, A242893, A242894, A242895.

A242893 Beginning with a centrally symmetric 'Sun' configuration of 9 rhombi with rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Original entry on oeis.org

9, 9, 9, 9, 18, 27, 27, 36

Offset: 1

Felix Fröhlich, May 25 2014

S. Weber, Fourier transform on quasiperiodic point sets (select '9-fold' with 'shift 1/n')

Cf. A242128, A242129, A242888, A242889, A242890, A242891, A242892, A242894, A242895.

A309118 Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.

Original entry on oeis.org

6, 6, 12, 18, 24, 24, 36, 30, 48, 36, 60, 42, 72, 48, 84, 54, 96, 60, 108, 66, 120, 72, 132, 78, 144, 84, 156, 90, 168, 96, 180, 102, 192, 108, 204, 114, 216, 120, 228, 126, 240, 132, 252, 138, 264, 144, 276, 150, 288, 156, 300, 162, 312, 168, 324, 174, 336

Offset: 1

Felix Fröhlich, Jul 13 2019

			See illustration in Fröhlich, 2019.

Colin Barker, Table of n, a(n) for n = 1..1000
Felix Fröhlich, Layers in the rhombille tiling, 2019.
Wikipedia, Rhombille tiling
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A008588, A008594.

Cf. A242128 (5-fold, Star), A242129 (5-fold, Sun), A242888 (7-fold, Star), A242889 (7-fold, Sun), A242890 (8-fold, Star), A242891 (8-fold, Sun), A242892 (9-fold, Star), A242893 (9-fold, Sun), A242894 (Kite and dart, Star), A242895 (Kite and dart, Sun).

I:=[6,6,12,18,24,24]; [n le 6 select I[n] else 2*Self(n-2)-Self(n-4): n in [1..60]]; // Vincenzo Librandi, Jul 16 2019

Join[{6, 6}, LinearRecurrence[{0, 2, 0, -1}, {12, 18, 24, 24}, 60]] (* Vincenzo Librandi, Jul 16 2019 *)

a(n) = if(n<3, 6, if(n%2==0, 6*((n+2)/2), 12*((n-1)/2)))

Vec(6*x*(1 + x + x^3 + x^4 - x^5) / ((1 - x)^2*(1 + x)^2) + O(x^40)) \\ Colin Barker, Jul 13 2019

a(2*n+1) = A008594(n).
a(2*n) = A008588(n+1) for n > 1.
From Colin Barker, Jul 13 2019: (Start)
G.f.: 6*x*(1 + x + x^3 + x^4 - x^5) / ((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4) for n>6.
(End)

cp's OEIS Frontend

A242129 Number of tiles added in each step if a centrally symmetric Penrose rhomb tiling, beginning with a central 'Sun' configuration, is successively extended in steps as defined below.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A242128 Beginning with adding a central 'Star' configuration of a centrally symmetric Penrose rhomb tiling as the first step, number of tiles that can be added to the free edges of the current tiling.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

A242888 Beginning with adding a central 'Star' configuration of a centrally symmetric rhomb tiling with 7-fold symmetry, number of tiles that can be added to the free edges of the tiling.

Views

Author

Keywords

Links

Crossrefs

A242894 Beginning with a central 'Star' configuration of a Penrose 'Kite' and 'Dart' tiling with rotational symmetry as the first step, number of tiles that can be added to the free edges of the current tiling.

Views

Author

Keywords

Comments

Links

Crossrefs

A242895 Beginning with a central 'Sun' configuration of a Penrose 'Kite' and 'Dart' tiling with rotational symmetry as the first step, number of tiles that can be added to the free edges of the current tiling.

Views

Author

Keywords

Comments

Links

Crossrefs

A242890 Beginning with a central 'Star' configuration of a centrally symmetric arrangement of 8 rhombi with 8-fold rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Views

Author

Keywords

Links

Crossrefs

A242891 Beginning with a centrally symmetric 'Sun' configuration of 8 rhombi with rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Views

Author

Keywords

Links

Crossrefs

A242892 Beginning with a centrally symmetric 'Star' configuration of 9 rhombi with rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Views

Author

Keywords

Links

Crossrefs

A242893 Beginning with a centrally symmetric 'Sun' configuration of 9 rhombi with rotational symmetry, number of tiles that can be added to the free edges of the tiling.

Views

Author

Keywords

Links

Crossrefs

A309118 Number of tiles added at iteration n when successively, layer by layer, building a symmetric patch of a rhombille tiling around a central star of six rhombs.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula