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, and 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
a(n) = Sum_{k=0..n} (-1)^k*2^(n-2*k)*binomial(n+1,2*k+1). - Gerry Martens, Sep 18 2022
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
KEYWORD
easy,sign
AUTHOR
Paul Barry, Oct 16 2004
STATUS
approved