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 A099456 Expansion of 1/(1 - 4*x + 5*x^2). 10
 1, 4, 11, 24, 41, 44, -29, -336, -1199, -3116, -6469, -10296, -8839, 16124, 108691, 354144, 873121, 1721764, 2521451, 1476984, -6699319, -34182196, -103232189, -242017776, -451910159, -597551756, -130656229, 2465133864 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Associated to the knot 9_44 by the modified Chebyshev transform A(x) -> (1/(1+x^2)^2)A(x/(1+x^2)). See A099457 and A099458. Imaginary part of (2+i)^n. - Gary W. Adamson, Apr 05 2008; Franklin T. Adams-Watters, Jan 06 2009 LINKS Seiichi Manyama, Table of n, a(n) for n = 0..1000 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. Dror Bar-Natan, The Rolfsen Knot Table Index entries for linear recurrences with constant coefficients, signature (4,-5). FORMULA a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-5)^k*4^(n-2k). E.g.f. (with offset 1): exp(x)^2*sin(x). - Zerinvary Lajos, Apr 06 2009 [corrected by Joerg Arndt, Apr 24 2011] a(n) = 4*a(n-1) - 5*a(n-2), a(0)=1, a(1)=4. - Vincenzo Librandi, Mar 22 2011 From Paul Curtz, Apr 24 2011: (Start) a(n) - a(n-4) = 40 * A118444(n); a(n) - a(n-2) = 10 * A139011(n). (End) a(n) = ((1+2*i)*(2-i)^n + (1-2*i)*(2+i)^n)/2. - Vaclav Kotesovec, Oct 09 2013 a(n) = ((2+i)^(n+1) - (2-i)^(n+1))/(2*i). Lim sup n->infinity |a(n)|/5^((n+1)/2) = 1. - Vaclav Kotesovec, Oct 09 2013 MAPLE seq(((2+I)^(n+1) - (2-I)^(n+1))/(2*I), n=0..30);  # James R. Buddenhagen, Dec 29 2017 MATHEMATICA CoefficientList[Series[1/(1-4*x+5*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 09 2013 *) Table[((1+2*I)*(2-I)^n + (1-2*I)*(2+I)^n)/2, {n, 0, 20}] (* Vaclav Kotesovec, Oct 09 2013 *) PROG (Sage) [lucas_number1(n, 4, 5) for n in range(1, 29)] # Zerinvary Lajos, Apr 22 2009 CROSSREFS Cf. A139011. - Franklin T. Adams-Watters, Jan 06 2009 Sequence in context: A008070 A008096 A008209 * A008069 A047950 A008259 Adjacent sequences:  A099453 A099454 A099455 * A099457 A099458 A099459 KEYWORD easy,sign AUTHOR Paul Barry, Oct 16 2004 STATUS approved

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Last modified June 24 07:56 EDT 2021. Contains 345416 sequences. (Running on oeis4.)