1,1

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)counts 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.

Wikipedia, T-square (fractal)

Good math, bad math, Geometric L-systems

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

G.f.: 4*x*(1-5*x^2+2*x^3+4*x^4)/((1-x)*(1-2*x)*(1-3*x)) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 21 2009]

a(n)=4+(92/27)*3^n-2*2^n-(56/9)*{C[2*(n-1),n-1] mod 2}-(8/3)*[C(n^2,n+2) mod 2], with n>=1 [From Paolo P. Lava (ppl(AT)spl.at), Mar 31 2009]

(PARI) a(n)=4*((n==1)+(n==2)*6+(n>=3)*(1-2^(n-1)+23*3^(n-3))) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 22 2009]

Cf. A000392. [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Mar 21 2009]

Sequence in context: A011915 A025220 A112742 this_sequence A069145 A005561 A061612

Adjacent sequences: A158491 A158492 A158493 this_sequence A158495 A158496 A158497

easy,nonn

Andrew V. Sutherland (drew(AT)math.mit.edu), Mar 20 2009

Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Oct 28 2009