A090018 - OEIS

%I #48 Oct 10 2022 02:46:53

%S 1,6,39,252,1629,10530,68067,439992,2844153,18384894,118841823,

%T 768205620,4965759189,32099171994,207492309531,1341251373168,

%U 8669985167601,56043665125110,362271946253463,2341762672896108

%N a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6.

%C From _Johannes W. Meijer_, Aug 09 2010: (Start)

%C 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)

%H Vincenzo Librandi, <a href="/A090018/b090018.txt">Table of n, a(n) for n = 0..200</a>

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (6,3).

%F a(n) = (3+2*sqrt(3))^n*(sqrt(3)/4+1/2) + (1/2-sqrt(3)/4)*(3-2*sqrt(3))^n.

%F a(n) = (-i*sqrt(3))^n * ChebyshevU(n, isqrt(3)), i^2=-1.

%F From _Johannes W. Meijer_, Aug 09 2010: (Start)

%F G.f.: 1/(1 - 6*x - 3*x^2).

%F Lim_{k->infinity} a(n+k)/a(k) = A141041(n) + A090018(n-1)*sqrt(12) for n >= 1.

%F Lim_{n->infinity} A141041(n)/A090018(n-1) = sqrt(12).

%F (End)

%F a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - _Philippe Deléham_, Nov 21 2011

%p a:= n-> (<<0|1>, <3|6>>^n. <<1,6>>)[1,1]:

%p seq(a(n), n=0..30);

%t Join[{a=1,b=6},Table[c=6*b+3*a;a=b;b=c,{n,100}]] (* _Vladimir Joseph Stephan Orlovsky_, Jan 16 2011 *)

%t LinearRecurrence[{6,3}, {1,6}, 41] (* _G. C. Greubel_, Oct 10 2022 *)

%o (Sage) [lucas_number1(n,6,-3) for n in range(1, 31)] # _Zerinvary Lajos_, Apr 24 2009

%o (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

%o (PARI) my(x='x+O('x^30)); Vec(1/(1-6*x-3*x^2)) \\ _G. C. Greubel_, Jan 24 2018

%Y Cf. A180028, A180032.

%Y 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).

%K nonn,easy

%O 0,2

%A _Paul Barry_, Nov 19 2003

%E Typo in Mathematica program corrected by _Vincenzo Librandi_, Nov 15 2011