A240917 a(n) = 2*3^(2*n) - 3*3^n + 1.

0, 10, 136, 1378, 12880, 117370, 1060696, 9559378, 86073760, 774781930, 6973391656, 62761587778, 564857478640, 5083726873690, 45753570561016, 411782221142578, 3706040248563520, 33354363011912650, 300189269431736776, 2701703431859199778

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 (fan pattern) 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*(A198643-1). The perimeter seems to converge to 10/6.

Links

• Kival Ngaokrajang, Illustration of triflake like fractal (fan pattern) for n = 0..3
• Wikipedia, n-flake,
• Index entries for linear recurrences with constant coefficients, signature (13,-39, 27).

Crossrefs

Cf. A198643, A240523 (pentaflake), A240671 (heptaflake), A240572 (octaflake), A240733 (nonaflake), A240734 (decaflake), A240840 (hendecaflake), A240735 (dodecaflake), A240841 (tridecaflake).

Programs

Maple

A240917:=n->2*3^(2*n) - 3*3^n + 1; seq(A240917(n), n=0..30); # Wesley Ivan Hurt, Apr 15 2014

Mathematica

Table[2*3^(2 n) - 3*3^n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 15 2014 *)

PARI

a(n)= 2*3^(2*n) - 3*3^n + 1
       for(n=0,100,print1(a(n),", "))

PARI

concat(0, Vec(-2*x*(3*x+5)/((x-1)*(3*x-1)*(9*x-1)) + O(x^100))) \\ Colin Barker, Apr 15 2014

Formula

a(n) = 2*A007742(A003462(n)).

a(n) = 9*(a(n-1) + 2*A048473(n-1)) + 1.

From Colin Barker, Apr 15 2014: (Start)

a(n) = 1-3^(1+n)+2*9^n.

a(n) = 13*a(n-1)-39*a(n-2)+27*a(n-3).

G.f.: -2*x*(3*x+5) / ((x-1)*(3*x-1)*(9*x-1)). (End).