A158494 Boundary area of the T-square fractal.

4, 24, 80, 248, 768, 2360, 7200, 21848, 66048, 199160, 599520, 1802648, 5416128, 16264760, 48827040, 146546648, 439771008, 1319575160, 3959249760, 11878797848, 35638490688, 106919666360, 320767387680, 962318940248, 2886990375168, 8661038234360

Offset: 1

Author

Andrew V. Sutherland, Mar 20 2009

Keywords

• nonn
• easy

Comments

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.

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Wikipedia, T-square (fractal)
• Good math, bad math, Geometric L-systems
• Index entries for linear recurrences with constant coefficients, signature (6,-11,6).

Crossrefs

Cf. A000392.

Programs

Mathematica

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 *)

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

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

Formula

a(1)=4, a(2)=24, a(3)=80; for n>3, a(n) = 3*a(n-1) + 2^n - 8.

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

From Colin Barker, May 22 2017: (Start)

a(n) = 4 - 2^(n+1) + 92*3^(n-3) for n>2.

a(n) = 6*a(n-1) - 11*a(n-2) + 6*a(n-3) for n>5. (End)

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009