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A090018 a(n) = 6*a(n-1) + 3*a(n-2) for n > 2, a(0)=1, a(1)=6. 7
1, 6, 39, 252, 1629, 10530, 68067, 439992, 2844153, 18384894, 118841823, 768205620, 4965759189, 32099171994, 207492309531, 1341251373168, 8669985167601, 56043665125110, 362271946253463, 2341762672896108 (list; graph; refs; listen; history; text; internal format)
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.

This sequence belongs to a family of sequences with g.f. = 1/(1 - 6*x - k*x^2). The members of this family that are red queen sequences are A090018 (k=3; this sequence), A135030 (k=2), A005668 (k=1), A000400 (k=0), A001109 (k=-1), A154244 (k=-2), A138395 (k=-3), A084326 (k=-4) and A003463 (k=-5). Other members of this family are A135032 (k=4), A015551 (k=5), A057089 (k=6), A015552 (k=7), A030192 (k=-6), A081179 (k=-7), A006516 (k=-8), A027471 (k=-9) and A106392 (k=-10).

(End)

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Index entries for sequences related to Chebyshev polynomials.

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) = U(n, isqrt(3))(-isqrt(3))^n, i^2=-1.

From Johannes W. Meijer, Aug 09 2010: (Start)

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

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

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

(End)

a(n) = Sum_{k=0..n} A099089(n,k)*3^k. - Philippe Deléham, 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 *)

PROG

(Sage) [lucas_number1(n, 6, -3) for n in xrange(1, 21)] # Zerinvary Lajos, Apr 24 2009

(MAGMA) I:=[1, 6]; [n le 2 select I[n] else 6*Self(n-1)+3*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Nov 15 2011

(PARI) x='x+O('x^30); Vec(1/(1-6*x-3*x^2)) \\ G. C. Greubel, Jan 24 2018

CROSSREFS

Sequence in context: A037683 A145664 A305289 * A238809 A006256 A052392

Adjacent sequences:  A090015 A090016 A090017 * A090019 A090020 A090021

KEYWORD

nonn,easy

AUTHOR

Paul Barry, Nov 19 2003

EXTENSIONS

Typo in Mathematica program corrected by Vincenzo Librandi, Nov 15 2011

STATUS

approved

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Last modified October 16 16:34 EDT 2019. Contains 328101 sequences. (Running on oeis4.)