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A090018
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a(n) = 6*a(n-1)+3*a(n-2) for n>2, a(0)=1, a(1)=6.
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6
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| Contribution from Johannes W. Meijer, Aug 09 2010: (Start)
The a(n) represent 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 GF(x) = 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)
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for sequences related to Chebyshev polynomials.
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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.
Contribution from Johannes W. Meijer, Aug 09 2010: (Start)
G.f.: 1/(1-6*x-3*x^2).
Limit(a(n+k)/a(k), k=infinity) = A141041(n) + A090018(n-1)*sqrt(12) for n>=1.
Limit(A141041(n)/A090018(n-1), n=infinity) = sqrt(12).
(End)
a(n) = Sum_{k, 0<=k<=n} A099089(n,k)*3^k. - Philippe DELEHAM, Nov 21 2011
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MAPLE
| a:= n-> (<<0|1>, <3|6>>^n. <<1, 6>>)[1, 1]:
seq (a(n), n=0..30);
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MATHEMATICA
| Join[{a=1, b=6}, Table[c=6*b+3*a; a=b; b=c, {n, 100}]] (*From Vladimir Joseph Stephan Orlovsky, Jan 16 2011*)
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PROG
| (Other) sage: [lucas_number1(n, 6, -3) for n in xrange(1, 21)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), 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
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CROSSREFS
| Sequence in context: A037592 A037683 A145664 * A006256 A052392 A199491
Adjacent sequences: A090015 A090016 A090017 * A090019 A090020 A090021
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KEYWORD
| nonn,easy
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AUTHOR
| Paul Barry (pbarry(AT)wit.ie), Nov 19 2003
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EXTENSIONS
| Typo in Mma program fixed by Vincenzo Librandi, Nov 15 2011
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