A277491 - OEIS

%I #32 Feb 26 2023 12:57:55

%S 1,12,120,1128,10344,93864,847848,7642920,68835432,619715496,

%T 5578225896,50207178792,451877192040,4066945060008,36602706866664,

%U 329425167106344,2964829725182568,26683480411545000,240151375243512552,2161362583350043176,19452264074784109416

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

%C 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).

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Koch_snowflake">Koch snowflake</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (13,-36).

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

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

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

%e 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.

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

%t Table[1/5*(8*9^n - 3*4^n), {n, 0, 20}] (* or *)

%t CoefficientList[Series[(1 - x)/((1 - 4 x) (1 - 9 x)), {x, 0, 20}], x] (* _Michael De Vlieger_, Nov 10 2016 *)

%t LinearRecurrence[{13,-36},{1,12},30] (* _Harvey P. Dale_, Feb 26 2023 *)

%o (PARI) Vec((1-x)/((1-4*x)*(1-9*x)) + O(x^30)) \\ _Colin Barker_, Oct 19 2016

%o (Magma) [(8*9^n-3*4^n)/5 : n in [0..30]]; // _Wesley Ivan Hurt_, Apr 11 2017

%Y Cf. A277492.

%K nonn,easy

%O 0,2

%A _Dintle N Kagiso_, Oct 17 2016