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A235643 Total number of sides of a tetraflake-like fractal after n iterations, a(1) = 16 (see comments).

%I A235643 #23 Feb 16 2025 08:33:21
%S A235643 16,68,296,1300,5728,25268,111512,492196,2172592,9590180,42332936,
%T A235643 186866356,824867584,3641141012,16072772984,70948650820,313182494032,
%U A235643 1382454408452,6102448992488,26937513095764,118907935627168,524885022092660,2316954583165784
%N A235643 Total number of sides of a tetraflake-like fractal after n iterations, a(1) = 16 (see comments).
%C A235643 Construction rule is same as for box and Vicsek fractals, but uses 6 boxes at initial stage (n = 1) and has only one symmetrical axis. The scale factor of these fractals is 1/3. The actual tetraflake fractals have scale factor of 1/2.
%C A235643 a(n) is the total number of sides at different lengths of a tetraflake-like fractal after n iterations. The perimeter (rounded down) is A235648(n). The total number of holes is A241271(n+1).
%H A235643 Kival Ngaokrajang, <a href="/A235643/a235643.pdf">Illustration of initial terms</a>
%H A235643 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BoxFractal.html">Box Fractal</a>
%H A235643 Wikipedia, <a href="http://en.wikipedia.org/wiki/N-flake">n-flake</a>
%H A235643 Wikipedia, <a href="http://en.wikipedia.org/wiki/Vicsek_fractal">Vicsek Fractal</a>
%H A235643 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6, -7).
%F A235643 Conjecture from _Colin Barker_, Apr 21 2014: (Start)
%F A235643 a(n) = sqrt(2)*((3-sqrt(2))^n*(-1+sqrt(2))+(1+sqrt(2))*(3+sqrt(2))^n).
%F A235643 a(n) = 6*a(n-1)-7*a(n-2).
%F A235643 G.f.: 4*x*(-7*x+4) / (7*x^2-6*x+1). (End)
%t A235643 LinearRecurrence[{6,-7},{16,68},30] (* _Harvey P. Dale_, Jun 14 2014 *)
%o A235643 (Small Basic)
%o A235643 a[0] = 10
%o A235643 a[1] = 16
%o A235643 For n = 2 To 20
%o A235643   t1 = a[n-1]*3
%o A235643   t2 = a[n-2]*2
%o A235643   t3 = 0
%o A235643   If n >= 3 then
%o A235643     for i = 3 to n
%o A235643       t3 = t3 + a[i-3]*2*Math.Power(3,n-i+1)
%o A235643     EndFor
%o A235643   EndIf
%o A235643   a[n] = t1 + t2 + t3
%o A235643   TextWindow.Write(a[n-1]+", ")
%o A235643 EndFor
%Y A235643 Cf. A241271, A235648.
%Y A235643 Cf. A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).
%Y A235643 Cf. A063628 (hexaflake).
%Y A235643 Cf. A240916, A240917 (triflake-like); A238777 (tetraflake-like).
%K A235643 nonn
%O A235643 1,1
%A A235643 _Kival Ngaokrajang_, Apr 20 2014
%E A235643 More terms from _Harvey P. Dale_, Jun 14 2014