A242129 -id:A242129 - cp's OEIS frontend

A242935 Number of forced tiles in the local empire of the eight possible vertex configurations in a Penrose rhomb tiling, in the order D, J, S, Q, S5, K, S3, S4.

0, 2, 5, 10, 25, 27, 43, 99

Offset: 1

Felix Fröhlich, May 27 2014

The sequence refers to Figures 3.5 - 3.11 on pp. 28-32 of Effinger-Dean's thesis. It has been decided that the numbers should be listed here in the OEIS in order of increasing size. - M. F. Hasler, Jun 04 2019

L. Effinger-Dean, The Empire Problem in Penrose Tilings, B.A. thesis, Williams college (Williamstown, MA), May 2006, Fig. 3.5 - 3.11, pp. 28-32.
Wikipedia, Penrose tiling.

Cf. A242128, A242129, A242894, A242895, A243884, A302176, A302841, A302842, A235857, A235858, A103906, A103907. - M. F. Hasler, Jun 04 2019

Terms reordered by M. F. Hasler, Jun 04 2019

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)

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A242935 Number of forced tiles in the local empire of the eight possible vertex configurations in a Penrose rhomb tiling, in the order D, J, S, Q, S5, K, S3, S4.

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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.

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