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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 April 4 05:34 EDT 2020. Contains 333212 sequences. (Running on oeis4.)