OFFSET
0,2
COMMENTS
a(n) is the number of holes shaped like six-pointed stars, in descending size, found in the cross-section, in the shape of a regular hexagon, of a Menger Sponge. - Albert Säfström, Jul 25 2018
REFERENCES
Murat Sahin and Elif Tan, Conditional (strong) divisibility sequences, Fib. Q., 56 (No. 1, 2018), 18-31.
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Albert Säfström, Illustration of regular hexagonal cross-section of Menger Sponge, with supporting triangular shape
Index entries for linear recurrences with constant coefficients, signature (9,-12)
FORMULA
G.f.: -(3*x-1)/(12*x^2-9*x+1). - Alois P. Heinz, Mar 10 2018
From Colin Barker, Mar 12 2018: (Start)
a(n) = 2^(-1-n)*((9-sqrt(33))^n*(-3+sqrt(33)) + (3+sqrt(33))*(9+sqrt(33))^n) / sqrt(33).
a(n) = 9*a(n-1) - 12*a(n-2) for n>1.
(End)
MAPLE
a:= n-> (<<0|1>, <-12|9>>^n. <<1, 6>>)[1, 1]:
seq(a(n), n=0..25); # Alois P. Heinz, Mar 10 2018
MATHEMATICA
CoefficientList[Series[-(3 x - 1)/(12 x^2 - 9 x + 1), {x, 0, 20}], x] (* Michael De Vlieger, Mar 10 2018 *)
LinearRecurrence[{9, -12}, {1, 6}, 30] (* Vincenzo Librandi, Mar 11 2018 *)
PROG
(Magma) I:=[1, 6]; [n le 2 select I[n] else 9*Self(n-1)-12*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Mar 11 2018
(PARI) Vec((1 - 3*x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ Colin Barker, Mar 12 2018
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Mar 10 2018
EXTENSIONS
More terms from Altug Alkan, Mar 10 2018
STATUS
approved