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

1,1

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).

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)

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 *)

Prog

(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

Crossrefs

Cf. A000392.

Sequence in context: A025220 A375092 A112742 * A209456 A069145 A264184

Adjacent sequences: A158491 A158492 A158493 * A158495 A158496 A158497

Keyword

nonn,easy

Author

Andrew V. Sutherland, Mar 20 2009

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

Status

approved