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A138229
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Expansion of (1-x)/(1-2x+6x^2).
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7
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1, 1, -4, -14, -4, 76, 176, -104, -1264, -1904, 3776, 18976, 15296, -83264, -258304, -17024, 1515776, 3133696, -2827264, -24456704, -31949824, 82840576, 357380096, 217716736, -1708847104, -4723994624, 805093376
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OFFSET
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0,3
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COMMENTS
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Binomial transform of [1, 0, -5, 0, 25, 0, -125, 0, 625, 0, ...]=: powers of -5 with interpolated zeros. - Philippe Deléham, Dec 02 2008
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LINKS
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FORMULA
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a(n) = 2*a(n-1) - 6*a(n-2), a(0)=1, a(1)=1.
a(n) = Sum_{k=0..n} A098158(n,k)*(-5)^(n-k). (End)
G.f.: G(0)/2, where G(k) = 1 + 1/(1 - x*(5*k+1)/(x*(5*k+6) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 26 2013
a(n) = real part of the quaternion (1 + i + 2*j)^n. - Peter Bala, Mar 29 2015
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MATHEMATICA
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CoefficientList[Series[(1-x)/(1-2x+6x^2), {x, 0, 30}], x] (* or *) LinearRecurrence[ {2, -6}, {1, 1}, 30] (* Harvey P. Dale, Feb 29 2012 *)
TrigExpand@Table[6^(n/2) Cos[n ArcTan[Sqrt[5]]], {n, 0, 20}] (* or *)
Table[Sum[(-5)^k Binomial[n, 2 k], {k, 0, n/2}], {n, 0, 20}] (* Vladimir Reshetnikov, Sep 20 2016 *)
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PROG
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(Sage) [lucas_number2(n, 2, 6)/2 for n in range(0, 28)] # Zerinvary Lajos, Jul 08 2008
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CROSSREFS
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KEYWORD
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easy,sign
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AUTHOR
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STATUS
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approved
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