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

0,2

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).

Links

Table of n, a(n) for n=0..20.

Wikipedia, Koch snowflake

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

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.

Example

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.

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 *)

Prog

(PARI) Vec((1-x)/((1-4*x)*(1-9*x)) + O(x^30)) \\ Colin Barker, Oct 19 2016
(Magma) [(8*9^n-3*4^n)/5 : n in [0..30]]; // Wesley Ivan Hurt, Apr 11 2017

Crossrefs

Cf. A277492.

Sequence in context: A008465 A291391 A115902 * A004332 A129329 A266393

Adjacent sequences: A277488 A277489 A277490 * A277492 A277493 A277494

Keyword

nonn,easy

Author

Dintle N Kagiso, Oct 17 2016

Status

approved