

A158494


Boundary area of the Tsquare fractal.


1



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
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OFFSET

1,1


COMMENTS

Consider the nth iteration of the Tsquare 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, Tsquare (fractal)
Good math, bad math, Geometric Lsystems
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(n1) + 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
a(n) = 4 + (92/27)*3^n  2*2^n(56/9)*(binomial(2*(n1),n1) mod 2)  (8/3)*(binomial(n^2,n+2) mod 2).  Paolo P. Lava, Mar 31 2009
From Colin Barker, May 22 2017: (Start)
a(n) = 4  2^(n+1) + 92*3^(n3) for n>2.
a(n) = 6*a(n1)  11*a(n2) + 6*a(n3) 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)*(12^(n1)+23*3^(n3))) \\ Jaume Oliver Lafont, Mar 22 2009
(PARI) Vec(4*x*(15*x^2+2*x^3+4*x^4) / ((1x)*(12*x)*(13*x)) + O(x^30)) \\ Colin Barker, May 22 2017


CROSSREFS

Cf. A000392.
Sequence in context: A250132 A025220 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



