OFFSET
0,2
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., SierpiĆski 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).
FORMULA
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).
MAPLE
MATHEMATICA
Table[2*3^(2 n) - 3*3^n + 1, {n, 0, 30}] (* Wesley Ivan Hurt, Apr 15 2014 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Apr 14 2014
STATUS
approved