A099456 - OEIS

%I #65 Oct 05 2022 13:08:42

%S 1,4,11,24,41,44,-29,-336,-1199,-3116,-6469,-10296,-8839,16124,108691,

%T 354144,873121,1721764,2521451,1476984,-6699319,-34182196,-103232189,

%U -242017776,-451910159,-597551756,-130656229,2465133864

%N Expansion of 1/(1 - 4*x + 5*x^2).

%C 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.

%C Imaginary part of (2+i)^n. - _Gary W. Adamson_, Apr 05 2008; _Franklin T. Adams-Watters_, Jan 06 2009

%H Seiichi Manyama, <a href="/A099456/b099456.txt">Table of n, a(n) for n = 0..1000</a>

%H Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, <a href="https://doi.org/10.1007/s00006-019-0969-9">On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis</a>, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.

%H Dror Bar-Natan, <a href="http://katlas.org/wiki/The_Rolfsen_Knot_Table">The Rolfsen Knot Table</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5).

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*(-5)^k*4^(n-2k).

%F E.g.f. (with offset 1): exp(x)^2*sin(x). - _Zerinvary Lajos_, Apr 06 2009 [corrected by _Joerg Arndt_, Apr 24 2011]

%F a(n) = 4*a(n-1) - 5*a(n-2), a(0)=1, a(1)=4. - _Vincenzo Librandi_, Mar 22 2011

%F From _Paul Curtz_, Apr 24 2011: (Start)

%F a(n) - a(n-4) = 40 * A118444(n);

%F a(n) - a(n-2) = 10 * A139011(n). (End)

%F a(n) = ((1+2*i)*(2-i)^n + (1-2*i)*(2+i)^n)/2. - _Vaclav Kotesovec_, Oct 09 2013

%F a(n) = ((2+i)^(n+1) - (2-i)^(n+1))/(2*i).

%F Lim sup n->infinity |a(n)|/5^((n+1)/2) = 1. - _Vaclav Kotesovec_, Oct 09 2013

%F a(n) = Sum_{k=0..n} (-1)^k*2^(n-2*k)*binomial(n+1,2*k+1). - _Gerry Martens_, Sep 18 2022

%p seq(((2+I)^(n+1) - (2-I)^(n+1))/(2*I),n=0..30);  # _James R. Buddenhagen_, Dec 29 2017

%t CoefficientList[Series[1/(1-4*x+5*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 09 2013 *)

%t Table[((1+2*I)*(2-I)^n + (1-2*I)*(2+I)^n)/2,{n,0,20}] (* _Vaclav Kotesovec_, Oct 09 2013 *)

%o (Sage) [lucas_number1(n,4,5) for n in range(1, 29)] # _Zerinvary Lajos_, Apr 22 2009

%Y Cf. A139011, A090131 (inv. bin. transf.)

%K easy,sign

%O 0,2

%A _Paul Barry_, Oct 16 2004