A277491 Number of triangles in the standard triangulation of the n-th approximation of the Koch snowflake fractal.

1, 12, 120, 1128, 10344, 93864, 847848, 7642920, 68835432, 619715496, 5578225896, 50207178792, 451877192040, 4066945060008, 36602706866664, 329425167106344, 2964829725182568, 26683480411545000, 240151375243512552, 2161362583350043176, 19452264074784109416

Offset: 0

Author

Dintle N Kagiso, Oct 17 2016

Keywords

• nonn
• easy

Comments

The rational function A_n = (a_0)/5 * (8 - 3(4/9)^n) where a_0=1 in the Wikipedia link below equals A_n = 1/9^n*a(n).

Examples

			a(1) = 9+3 = 12, because an equilateral triangle can be cut up into 9 triangles with side length one-third and 3 further triangles are stacked onto the three central side pieces.
		

Links

• Wikipedia, Koch snowflake
• Index entries for linear recurrences with constant coefficients, signature (13,-36).

Crossrefs

Cf. A277492.

Programs

Magma

[(8*9^n-3*4^n)/5 : n in [0..30]]; // Wesley Ivan Hurt, Apr 11 2017

Maple

L:=[1,12]: for k from 3 to 34 do: L:=[op(L),13*L[k-1]-36*L[k-2]]: od: print(L);

Mathematica

Table[1/5*(8*9^n - 3*4^n), {n, 0, 20}] (* or *)
CoefficientList[Series[(1 - x)/((1 - 4 x) (1 - 9 x)), {x, 0, 20}], x] (* Michael De Vlieger, Nov 10 2016 *)
LinearRecurrence[{13,-36},{1,12},30] (* Harvey P. Dale, Feb 26 2023 *)

PARI

Vec((1-x)/((1-4*x)*(1-9*x)) + O(x^30)) \\ Colin Barker, Oct 19 2016

Formula

G.f.: (1-x) / ((1-4*x)*(1-9*x)).

a(n) = 13*a(n-1) - 36*a(n-2) for n>1, a(0)=1, a(1)=12.

a(n) = (8*9^n-3*4^n)/5.