OFFSET
0,2
COMMENTS
From Johannes W. Meijer, Aug 09 2010: (Start)
a(n) represents the number of n-move routes of a fairy chess piece starting in a given corner or side square on a 3 X 3 chessboard. This fairy chess piece behaves like a white queen on the eight side and corner squares but on the central square the queen explodes with fury and turns into a red queen, see A180032. The central square leads to A180028. (End)
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (6,3).
FORMULA
a(n) = (3+2*sqrt(3))^n*(sqrt(3)/4+1/2) + (1/2-sqrt(3)/4)*(3-2*sqrt(3))^n.
a(n) = (-i*sqrt(3))^n * ChebyshevU(n, isqrt(3)), i^2=-1.
From Johannes W. Meijer, Aug 09 2010: (Start)
G.f.: 1/(1 - 6*x - 3*x^2).
(End)
a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - Philippe Del�ham, Nov 21 2011
MAPLE
a:= n-> (<<0|1>, <3|6>>^n. <<1, 6>>)[1, 1]:
seq(a(n), n=0..30);
MATHEMATICA
Join[{a=1, b=6}, Table[c=6*b+3*a; a=b; b=c, {n, 100}]] (* Vladimir Joseph Stephan Orlovsky, Jan 16 2011 *)
LinearRecurrence[{6, 3}, {1, 6}, 41] (* G. C. Greubel, Oct 10 2022 *)
PROG
(Sage) [lucas_number1(n, 6, -3) for n in range(1, 31)] # Zerinvary Lajos, Apr 24 2009
(Magma) [n le 2 select 6^(n-1) else 6*Self(n-1)+3*Self(n-2): n in [1..30]]; // Vincenzo Librandi, Nov 15 2011
(PARI) my(x='x+O('x^30)); Vec(1/(1-6*x-3*x^2)) \\ G. C. Greubel, Jan 24 2018
CROSSREFS
Sequences with g.f. of the form 1/(1 - 6*x - k*x^2): A106392 (k=-10), A027471 (k=-9), A006516 (k=-8), A081179 (k=-7), A030192 (k=-6), A003463 (k=-5), A084326 (k=-4), A138395 (k=-3), A154244 (k=-2), A001109 (k=-1), A000400 (k=0), A005668 (k=1), A135030 (k=2), this sequence (k=3), A135032 (k=4), A015551 (k=5), A057089 (k=6), A015552 (k=7), A189800 (k=8), A189801 (k=9), A190005 (k=10), A015553 (k=11).
KEYWORD
nonn,easy
AUTHOR
Paul Barry, Nov 19 2003
EXTENSIONS
Typo in Mathematica program corrected by Vincenzo Librandi, Nov 15 2011
STATUS
approved