OFFSET
0,2
COMMENTS
From Greg Dresden, Jun 25 2022: (Start)
a(n) is the number of ways to tile, with squares and dominoes, a 2 X n board with one extra space at the end. Here is the board for n=3:
_____
|_|_|_|_
|_|_|_|_|
and here is one of the a(3)=32 possible tilings of this board:
_____
| |_|_|_
|_|_|___|
(End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Richard M. Low and Ardak Kapbasov, Non-Attacking Bishop and King Positions on Regular and Cylindrical Chessboards, Journal of Integer Sequences, Vol. 20 (2017), Article 17.6.1, Table 4.
N. J. A. Sloane Notes on A030186 and A033505
Index entries for linear recurrences with constant coefficients, signature (3,1,-1).
FORMULA
a(n) = 3*a(n-1) + a(n-2) - a(n-3). - Greg Dresden, Aug 16 2018
MAPLE
seq(coeff(series(1/(1-3*x-x^2+x^3), x, n+1), x, n), n = 0..30); # G. C. Greubel, Oct 14 2019
MATHEMATICA
CoefficientList[Series[1/(1-3x-x^2+x^3), {x, 0, 30}], x] (* or *) LinearRecurrence[{3, 1, -1}, {1, 3, 10}, 30] (* Vincenzo Librandi, Aug 17 2018 *)
PROG
(Magma) I:=[1, 3, 10]; [n le 3 select I[n] else 3*Self(n-1)+Self(n-2)-Self(n-3): n in [1..30]]; // Vincenzo Librandi, Aug 17 2018
(PARI) my(x='x+O('x^30)); Vec(1/(1-3*x-x^2+x^3)) \\ G. C. Greubel, Oct 14 2019
(Sage)
def A033505_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P(1/(1-3*x-x^2+x^3)).list()
A033505_list(30) # G. C. Greubel, Oct 14 2019
(GAP) a:=[1, 3, 10];; for n in [4..30] do a[n]:=3*a[n-1]+a[n-2]-a[n-3]; od; a; # G. C. Greubel, Oct 14 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
N. J. A. Sloane, Feb 13 2002
STATUS
approved