A240916 a(n) = 6*a(n-1) + 2*2^(n-1) - 2 for n > 2, a(0) = a(1) = 0, a(2) = 3.

0, 0, 3, 24, 158, 978, 5930, 35706, 214490, 1287450, 7725722, 46356378, 278142362, 1668862362, 10013190554, 60079176090, 360475122074, 2162850863514, 12977105443226, 77862633183642, 467175800150426, 2803054802999706, 16818328822192538

Offset: 0

Author

Kival Ngaokrajang, Apr 14 2014

Keywords

• nonn
• easy

Comments

a(n) is the total number of holes of a triflake-like fractal (Mitsubishi logo) after n iterations. The scale factor for this case is 1/3, but for the actual triflake case, it is 1/2, i.e., Sierpiński triangle. The total number of sides is 3*A000302(n). The perimeter (rounded down) is A064628(n).

Links

• Kival Ngaokrajang, Illustration of triflake like fractal (Mitsubishi logo) for n = 0..3
• Wikipedia, n-flake
• Index entries for linear recurrences with constant coefficients, signature (9,-20,12).

Crossrefs

Cf. A000302, A064628, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).

Programs

Mathematica

Join[{0,0},LinearRecurrence[{9,-20,12},{3,24,158},30]] (* Harvey P. Dale, Jan 31 2015 *)

PARI

{a(n)=if(n<=0, 0, if(n<2, 0, if(n<3, 3, 6*a(n-1)+2*2^(n-1)-2)))}
  for(n=0,100,print1(a(n),", "))

PARI

concat([0,0], Vec(-x^2*(2*x^2-3*x+3)/((x-1)*(2*x-1)*(6*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014

Formula

From Colin Barker, Apr 15 2014: (Start)

a(n) = (72-45*2^(1+n)+23*6^n)/180 for n>1.

a(n) = 9*a(n-1)-20*a(n-2)+12*a(n-3) for n>4.

G.f.: -x^2*(2*x^2-3*x+3) / ((x-1)*(2*x-1)*(6*x-1)). (End).