A104565 - OEIS

%I #39 Mar 24 2023 15:23:03

%S 1,-2,3,-2,-6,28,-61,54,158,-860,2062,-2004,-5804,33720,-84509,86054,

%T 247862,-1492908,3838298,-4019452,-11537556,71101832,-185868978,

%U 198310460,567902572,-3555617432,9404104764,-10168382696,-29069700056,184127171952,-491229517661

%N Reversion of Pell numbers A000129(n+1).

%H Alois P. Heinz, <a href="/A104565/b104565.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: (sqrt(1+4*x+8*x^2)-1-2*x)/(2*x^2).

%F a(n) = sum{k=0..floor(n/2), binomial(n, 2k)*C(k)*(-1)^(n-k)2^(n-2k)}, where C(n) is A000108. - _Paul Barry_, May 16 2005

%F G.f. 1/G(0) where G(k)=  1 + 2*x + x^2/G(k+1); (continued fraction, 1-step). - _Sergei N. Gladkovskii_, Aug 10 2012

%F G.f.: (2/W(0)-1)/x  where W(k)=  1 + 1/(1 + 2*x/(1 + 2*x/W(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Sep 21 2012

%F D-finite with recurrence (n+2)*a(n) +2*(2*n+1)*a(n-1) +8*(n-1)*a(n-2)=0. - _R. J. Mathar_, Nov 09 2012

%F G.f.: G(0)/x^2 - 1/x - 1/x^2 where G(k)=  1 + 2*x/(1 + 1/(1 + 2*x/G(k+1))); (continued fraction, 3-step). - _Sergei N. Gladkovskii_, Nov 23 2012

%F G.f.: 1/(x^2*Q(0)) - 1/(x^2) - 1/x, where Q(k)= 1 - (4*k+1)*x*(1+2*x)/(k+1 - x*(1+2*x)*(2*k+2)*(4*k+3)/(2*x*(1+2*x)*(4*k+3) - (2*k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 15 2013

%F Lim sup n->infinity |a(n)|^(1/n) = 2*sqrt(2). - _Vaclav Kotesovec_, Feb 08 2014

%F a(n) = (-2)^n*hypergeom([1/2-n/2,-n/2], [2], -1). - _Vladimir Reshetnikov_, Nov 07 2015

%p a:= proc(n) a(n):= `if`(n<2, 1-3*n,

%p       ((8-8*n)*a(n-2)-(4*n+2)*a(n-1))/(n+2))

%p     end:

%p seq (a(n), n=0..40);  # _Alois P. Heinz_, Nov 09 2012

%t CoefficientList[Series[(Sqrt[1+4*x+8*x^2]-1-2*x)/(2*x^2), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 08 2014 *)

%t Table[(-2)^n Hypergeometric2F1[1/2-n/2, -n/2, 2, -1], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 07 2015 *)

%o (Sage)

%o def A104565_list(n):  # n>=1

%o     T = [0]*(n+1); R = [1]

%o     for m in (1..n-1):

%o         a,b,c = 1,0,0

%o         for k in range(m,-1,-1):

%o             r = a - 2*b - c

%o             if k < m : T[k+2] = u;

%o             a,b,c = T[k-1],a,b

%o             u = r

%o         T[1] = u; R.append(u)

%o     return R

%o A104565_list(30)  # _Peter Luschny_, Nov 01 2012

%K easy,sign

%O 0,2

%A _Paul Barry_, Mar 15 2005