%I A052530
%S A052530 0,2,8,30,112,418,1560,5822,21728,81090,302632,1129438,4215120,15731042,
%T A052530 58709048,219105150,817711552,3051741058,11389252680,42505269662,
%U A052530 158631825968,592022034210,2209456310872,8245803209278,30773756526240
%N A052530 a(0)=0, a(1)=2; for n>=2, a(n)=4*a(n-1)-a(n-2).
%C A052530 a(n-1) and a(n+1) are the solutions for c if b=a(n) in (b^2+c^2)/(b*c+1)=4 
               and there are no other pairs of solutions apart from consecutive 
               pairs of terms in this sequence. Cf. A061167. - Henry Bottomley (se16(AT)btinternet.com), 
               Apr 18 2001
%C A052530 a(n)^2 for n >= 1 gives solutions to A007913(3x+4)=A007913(x) - Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002
%C A052530 Also a(n+2)=4a(n+1)-a(n); also for all n, a(n)=[(3-2*sqrt(3))/3]*[(2-sqrt(3)]^n 
               + [(3+2*sqrt(3))/3]*[(2+sqrt(3)]^n [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Jan 04 2009]
%C A052530 For all n we have: [[(a(n)]^2 + [(a(n+1)]^2]/[a(n)*a(n+1)+1]=4 [From 
               Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Jan 04 2009]
%H A052530 T. D. Noe, <a href="b052530.txt">Table of n, a(n) for n=0..200</a>
%H A052530 J.-P. Ehrmann et al., <a href="http://forumgeom.fau.edu/POLYA/ProblemCenter/
               POLYA002.html">Problem POLYA002</a>, Integer pairs (x,y) for which 
               (x^2+y^2)/(1+pxy) is an integer.
%H A052530 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=460">
               Encyclopedia of Combinatorial Structures 460</a>
%H A052530 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A052530 G.f.: 2*x/(1-4*x+x^2)
%F A052530 Invert transform of even numbers: a(n)=2*Sum_{k=1..n} k*a(n-k). - Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Apr 27 2001
%F A052530 a(n) = Sum(-(1/3)*(-1+2*_alpha)*_alpha^(-1-n), _alpha=RootOf(1-4*_Z+_Z^2)), 
               i.e., a(n) = [ [(2+Sqrt(3)^n) - (2-Sqrt(3)^n)] - [(2+Sqrt(3)^(n-1)) 
               - (2-Sqrt(3)^(n-1))] + [(2+Sqrt(3)^(n-2)) - (2-Sqrt(3)^(n-2))] ] 
               / (3*Sqrt(3)) - Gregory V. Richardson (omomom(AT)hotmail.com), Oct 
               06 2002
%F A052530 For all elements n of the sequence, 3*n^2 + 4 is a perfect square. Lim. 
               a(n)/a(n-1) = 2 + Sqrt(3). - Gregory V. Richardson (omomom(AT)hotmail.com), 
               Oct 06 2002
%F A052530 a(n) = A071954(n) - 2.
%F A052530 a(n) = (2*Sinh[2n*ArcSinh[1/Sqrt[2]]])/Sqrt[3] - Herbert Kociemba (kociemba(AT)t-online.de), 
               Apr 24 2008
%F A052530 a(n)=2*A001353(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), 
               Oct 26 2009]
%p A052530 spec := [S,{S=Sequence(Prod(Union(Z,Z),Sequence(Z),Sequence(Z)))},unlabeled]: 
               seq(combstruct[count](spec, size=n), n=0..20);
%t A052530 a[0]:=0; a[1] = c; a[n_] := a[n] = p*c^2*a[n - 1] - a[n - 2]; p = 1; 
               c = 2; Table[ a[n], {n, 0, 20} ]
%t A052530 a[n_]:=(2*Sinh[2n*ArcCsch[Sqrt[2]]])/Sqrt[3];Table[a[n],{n, 0, 20}]//
               FullSimplify - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 
               2008
%o A052530 (PARI): { polya002(p,c,m) = local(v,w,j,a); w=0; print1(w,", "); v=c; 
               print1(v,", "); j=1; while(j<=m,a=p*c^2*v-w; print1(a,", "); w=v; 
               v=a; j++) } polya002(1,2,25)
%Y A052530 Cf. A007913, A003699.
%Y A052530 Sequence in context: A010749 A127865 A077839 this_sequence A162551 A073663 
               A155116
%Y A052530 Adjacent sequences: A052527 A052528 A052529 this_sequence A052531 A052532 
               A052533
%K A052530 easy,nonn,nice
%O A052530 0,2
%A A052530 encyclopedia(AT)pommard.inria.fr, Jan 25 2000
%E A052530 More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000
%E A052530 Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 11 2006
%E A052530 Max Alekseyev (maxale(AT)gmail.com) changed a(0) to 0 and revised the 
               entry accordingly, Nov 15 2007
%E A052530 Signs in definition corrected by John W. Layman, Nov 20 2007