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.

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

Author

Felix Fröhlich, Jul 13 2019

Keywords

• nonn
• easy

Examples

			See illustration in Fröhlich, 2019.
		

Links

• 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).

Crossrefs

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).

Programs

Magma

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

Mathematica

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

PARI

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

PARI

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

Formula

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)