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%I #22 Jan 03 2024 23:47:25
%S 4,24,80,248,768,2360,7200,21848,66048,199160,599520,1802648,5416128,
%T 16264760,48827040,146546648,439771008,1319575160,3959249760,
%U 11878797848,35638490688,106919666360,320767387680,962318940248,2886990375168,8661038234360
%N Boundary area of the T-square fractal.
%C Consider the n-th iteration of the T-square fractal (as defined in the links) drawn on an integer lattice scaled so that the shortest edge on the boundary of the fractal has unit length a(n)gives the number of lattice squares in the unshaded region that are adjacent to a square in the shaded region. For n=1 there is a single shaded square and a(1) counts the 4 adjacent unshaded squares. Proposed by Simone Severini.
%H Colin Barker, <a href="/A158494/b158494.txt">Table of n, a(n) for n = 1..1000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/T-square_(fractal)">T-square (fractal)</a>
%H Good math, bad math, <a href="http://scienceblogs.com/goodmath/2007/08/geometric_lsystems.php">Geometric L-systems</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-11,6).
%F a(1)=4, a(2)=24, a(3)=80; for n>3, a(n) = 3*a(n-1) + 2^n - 8.
%F G.f.: 4*x*(1 - 5*x^2 + 2*x^3 + 4*x^4) / ((1 - x)*(1 - 2*x)*(1 - 3*x)). - _Jaume Oliver Lafont_, Mar 21 2009
%F From _Colin Barker_, May 22 2017: (Start)
%F a(n) = 4 - 2^(n+1) + 92*3^(n-3) for n>2.
%F a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>5. (End)
%t CoefficientList[Series[4*(1 - 5*x^2 + 2*x^3 + 4*x^4)/((1 - x)*(1 - 2*x)*(1 - 3*x)), {x, 0, 30}], x] (* _Wesley Ivan Hurt_, Jan 20 2017 *)
%o (PARI) a(n)=4*((n==1)+(n==2)*6+(n>=3)*(1-2^(n-1)+23*3^(n-3))) \\ _Jaume Oliver Lafont_, Mar 22 2009
%o (PARI) Vec(4*x*(1-5*x^2+2*x^3+4*x^4) / ((1-x)*(1-2*x)*(1-3*x)) + O(x^30)) \\ _Colin Barker_, May 22 2017
%Y Cf. A000392.
%K nonn,easy
%O 1,1
%A _Andrew V. Sutherland_, Mar 20 2009
%E Edited by _Charles R Greathouse IV_, Oct 28 2009
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