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 A138229 Expansion of (1-x)/(1-2x+6x^2). 7
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Binomial transform of [1, 0, -5, 0, 25, 0, -125, 0, 625, 0, ...]=: powers of -5 with interpolated zeros . - Philippe Deléham, Dec 02 2008 LINKS Michael De Vlieger, Table of n, a(n) for n = 0..2571 Beata Bajorska-Harapińska, Barbara Smoleń, Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54. Index entries for linear recurrences with constant coefficients, signature (2,-6). FORMULA a(n) = 2*a(n-1)-6*a(n-2), a(0)=1, a(1)=1. a(n) = Sum_{k, 0<=k<=n}A098158(n,k)*(-5)^(n-k). - Philippe Deléham, Nov 14 2008 a(n) = Sum_{k, 0<=k<=n}A124182(n,k)*(-6)^(n-k). - Philippe Deléham, Nov 15 2008 a(n) = (1/2)*{[1-I*sqrt(5)]^n+[1+I*sqrt(5)]^n}, with n>=0 and I=sqrt(-1). - Paolo P. Lava, Nov 18 2008 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 MATHEMATICA 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 *) PROG (Sage) [lucas_number2(n, 2, 6)/2 for n in xrange(0, 28)] # Zerinvary Lajos, Jul 08 2008 CROSSREFS Cf. A088139. Sequence in context: A003117 A239465 A156985 * A131702 A276826 A029661 Adjacent sequences:  A138226 A138227 A138228 * A138230 A138231 A138232 KEYWORD easy,sign AUTHOR Paul Barry, Mar 06 2008 STATUS approved

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Last modified September 17 19:02 EDT 2019. Contains 327137 sequences. (Running on oeis4.)