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One third of the number of edges in the metrically regular triangulation of the n-th approximation of the Koch snowflake fractal.
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%I #13 Sep 22 2019 15:09:06

%S 1,8,68,596,5300,47444,425972,3829652,34450484,309988820,2789637236,

%T 25105686548,225946984628,2033506084436,18301487651060,

%U 164713120424084,1482417010074932,13341748795707092,120075721981494644,1080681429113975060

%N One third of the number of edges in the metrically regular triangulation of the n-th approximation of the Koch snowflake fractal.

%C A triangulation is metrically regular if all its triangles are congruent.

%C Three times the n-th sequence entry a(n) is the number of edges of the n-th approximation of the Koch snowflake fractal.

%H Dintle N Kagiso, <a href="/A277492/a277492.pdf">triangulation of snowflake</a>

%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 a(n+1) = (1/5) * (4*9^n +4^n) for all n > 0.

%F a(1)=1, a(2)=8, a(3)=68, a(n) = 13*a(n-1)-36*a(n-2) for n > 2.

%F G.f.: (1-5*x)/((1-4*x)*(1-9*x)).

%e a(1)=1, and there are three edges in a triangle. a(2)=8 and there are 24 edges in the second approximation of the Koch fractal.

%p L:=[1,8]: for k from 3 to 30 do: L:=[op(L),13*L[k-1]-36*L[k-2]]: od: print(L);

%t CoefficientList[Series[(1 - 5 x)/((1 - 4 x) (1 - 9 x)), {x, 0, 19}], x] (* or *) Table[(1/5) (4*9^# + 4^#) &[n + 1], {n, -1, 19}] (* _Michael De Vlieger_, Feb 18 2017 *)

%t LinearRecurrence[{13,-36},{1,8},30] (* _Harvey P. Dale_, Sep 22 2019 *)

%Y Cf. A277491, A277492.

%K nonn,easy

%O 1,2

%A _Dintle N Kagiso_, Feb 18 2017