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A240917 a(n) = 2*3^(2*n) - 3*3^n + 1.

%I A240917 #28 Feb 09 2024 12:39:12
%S A240917 0,10,136,1378,12880,117370,1060696,9559378,86073760,774781930,
%T A240917 6973391656,62761587778,564857478640,5083726873690,45753570561016,
%U A240917 411782221142578,3706040248563520,33354363011912650,300189269431736776,2701703431859199778
%N A240917 a(n) = 2*3^(2*n) - 3*3^n + 1.
%C A240917 a(n) is the total number of holes of a triflake-like fractal (fan pattern) after n iterations. The scale factor for this case is 1/3, but for the actual triflake case, it is 1/2, i.e., Sierpiński triangle. The total number of sides is 3*(A198643-1). The perimeter seems to converge to 10/6.
%H A240917 Kival Ngaokrajang, <a href="/A240917/a240917_2.pdf">Illustration of triflake like fractal (fan pattern) for n = 0..3</a>
%H A240917 Wikipedia, <a href="http://en.wikipedia.org/wiki/N-flake">n-flake,</a>
%H A240917 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (13,-39, 27).
%F A240917 a(n) = 2*A007742(A003462(n)).
%F A240917 a(n) = 9*(a(n-1) + 2*A048473(n-1)) + 1.
%F A240917 From _Colin Barker_, Apr 15 2014: (Start)
%F A240917 a(n) = 1-3^(1+n)+2*9^n.
%F A240917 a(n) = 13*a(n-1)-39*a(n-2)+27*a(n-3).
%F A240917 G.f.: -2*x*(3*x+5) / ((x-1)*(3*x-1)*(9*x-1)). (End).
%p A240917 A240917:=n->2*3^(2*n) - 3*3^n + 1; seq(A240917(n), n=0..30); # _Wesley Ivan Hurt_, Apr 15 2014
%t A240917 Table[2*3^(2 n) - 3*3^n + 1, {n, 0, 30}] (* _Wesley Ivan Hurt_, Apr 15 2014 *)
%o A240917 (PARI) a(n)= 2*3^(2*n) - 3*3^n + 1
%o A240917        for(n=0,100,print1(a(n),", "))
%o A240917 (PARI) concat(0, Vec(-2*x*(3*x+5)/((x-1)*(3*x-1)*(9*x-1)) + O(x^100))) \\ _Colin Barker_, Apr 15 2014
%Y A240917 Cf. A198643, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).
%K A240917 nonn,easy
%O A240917 0,2
%A A240917 _Kival Ngaokrajang_, Apr 14 2014