A099087 - OEIS

%I #65 Sep 08 2022 08:45:15

%S 1,2,2,0,-4,-8,-8,0,16,32,32,0,-64,-128,-128,0,256,512,512,0,-1024,

%T -2048,-2048,0,4096,8192,8192,0,-16384,-32768,-32768,0,65536,131072,

%U 131072,0,-262144,-524288,-524288,0,1048576,2097152,2097152,0,-4194304,-8388608,-8388608,0,16777216

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

%C Yet another variation on A009545.

%C Row sums of Krawtchouk triangle A098593. Partial sums of e.g.f. exp(x)cos(x), or 2^(n/2)cos(Pi*n/2). See A009116.

%C Binomial transform of A057077. - _R. J. Mathar_, Nov 04 2008

%C Partial sums of A146559. - _Philippe Deléham_, Dec 01 2008

%C Pisano period lengths: 1, 1, 8, 1, 4, 8, 24, 1, 24, 4, 40, 8, 12, 24, 8, 1, 16, 24, 72, 4, ... - _R. J. Mathar_, Aug 10 2012

%C Also the inverse Catalan transform of A000079. - _Arkadiusz Wesolowski_, Oct 26 2012

%H Seiichi Manyama, <a href="/A099087/b099087.txt">Table of n, a(n) for n = 0..5000</a>

%H Karl Dilcher and Maciej Ulas, <a href="https://arxiv.org/abs/2008.13475">Divisibility and Arithmetic Properties of a Class of Sparse Polynomials</a>, arXiv:2008.13475 [math.NT], 2020. See Table 1, 1st column, p. 3.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2).

%F E.g.f.: exp(x)*(cos(x) + sin(x)).

%F a(n) = 2^(n/2)*(cos(Pi*n/4) + sin(Pi*n/4)).

%F a(n) = Sum_{k=0..n} Sum_{i=0..k} binomial(n-k, k-i)*binomial(n, i) *(-1)^(k-i).

%F a(n) = 2*(a(n-1) - a(n-2)).

%F From _R. J. Mathar_, Apr 18 2008: (Start)

%F a(n) = (1-i)^(n-1) + (1+i)^(n-1) where i=sqrt(-1).

%F a(n) = 2 Sum_{k=0..(n-1)/2} (-1)^k*binomial(n-1,2k) if n>0. (End)

%F a(n) = Sum_{k=0..n} A109466(n,k)*2^k. - _Philippe Deléham_, Oct 28 2008

%F E.g.f.: (cos(x)+sin(x))*exp(x) = G(0); G(k)=1+2*x/(4*k+1-x*(4*k+1)/(2*(2*k+1)+x-2*(x^2)*(2*k+1)/((x^2)-(2*k+2)*(4*k+3)/G(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, Nov 26 2011

%F G.f.: U(0) where U(k)= 1 + x*(k+3) - x*(k+1)/U(k+1) ; (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Oct 10 2012

%F a(n) = Re((1+i)^n) + Im((1+i)^n) where i = sqrt(-1) = A146559(n) + A009545(n). - _Philippe Deléham_, Feb 13 2013

%F a(n) = Sum_{j=0..n} binomial(n, j)*(-1)^binomial(j, 2); this is the case m=2 and z=-1 of f(m,n)(z) = Sum_{j=0..n} binomial(n, j)*z^binomial(j, m). See Dilcher and Ulas. - _Michel Marcus_, Sep 01 2020

%t CoefficientList[Series[1/(1 -2x +2x^2), {x, 0, 50}], x] (* _Michael De Vlieger_, Dec 24 2015 *)

%o (Sage) [lucas_number1(n,2,2) for n in range(1, 50)] # _Zerinvary Lajos_, Apr 23 2009

%o (PARI) x='x+O('x^50); Vec(1/(1-2*x+2*x^2)) \\ _Altug Alkan_, Dec 24 2015

%o (Magma) I:=[1,2]; [n le 2 select I[n] else 2*(Self(n-1) - Self(n-2)): n in [1..50]]; // _G. C. Greubel_, Mar 16 2019

%o (GAP) a:=[1,2];; for n in [3..50] do a[n]:=2*a[n-1]-2*a[n-2]; od; a; # _G. C. Greubel_, Mar 16 2019

%Y Cf. A009545, A146559.

%K easy,sign

%O 0,2

%A _Paul Barry_, Sep 24 2004

%E Signs added by _N. J. A. Sloane_, Nov 14 2006