OFFSET
0,1
COMMENTS
a(n) is the total number of holes at n iterations of a fractal starting with a pattern of 15 boxes and 2 nonsquare holes (see illustration in Links field).
LINKS
Colin Barker, Table of n, a(n) for n = 0..849
Kival Ngaokrajang, Illustration of initial terms
Index entries for linear recurrences with constant coefficients, signature (19, -63, 45).
FORMULA
a(0) = 2, a(n) = 15*a(n-1) + 2 + 16*3^(n-1) for n > 0.
a(n) = 2*15^n + 2*Sum_{i=0..n-1} 15^i*(1 + 8*3^(n-1-i)).
G.f.: 2*(8*x^2-5*x-1) / ((x-1)*(3*x-1)*(15*x-1)). - Colin Barker, Aug 01 2015
a(n) = 19*a(n-1) - 63*a(n-2) + 45*a(n-3) for n>2. - Colin Barker, Aug 20 2015
MATHEMATICA
Table[(- 3 - 28 3^n + 73 15^n)/21, {n, 0, 20}] (* Vincenzo Librandi, Aug 22 2015 *)
LinearRecurrence[{19, -63, 45}, {2, 48, 770}, 20] (* Harvey P. Dale, Apr 18 2020 *)
PROG
(PARI) {for (n=0, 100, s=0; for (i=0, n-1, s=s+2*15^i*(1+8*3^(n-1-i))); a=s+2*15^n; print1(a, ", "))}
(PARI) Vec(2*(8*x^2-5*x-1)/((x-1)*(3*x-1)*(15*x-1)) + O(x^20)) \\ Colin Barker, Aug 01 2015
(Magma) [(-3-28*3^n+73*15^n)/21: n in [0..25]]; // Vincenzo Librandi, Aug 22 2015
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Kival Ngaokrajang, Aug 01 2015
EXTENSIONS
Condensed title by Colin Barker, Aug 01 2015
STATUS
approved