0,2

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

a(n)^2 for n >= 1 gives solutions to A007913(3x+4)=A007913(x) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 07 2002

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]

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]

T. D. Noe, Table of n, a(n) for n=0..200

J.-P. Ehrmann et al., Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer.

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 460

N. J. A. Sloane, Transforms

G.f.: 2*x/(1-4*x+x^2)

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

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

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

a(n) = A071954(n) - 2.

a(n) = (2*Sinh[2n*ArcSinh[1/Sqrt[2]]])/Sqrt[3] - Herbert Kociemba (kociemba(AT)t-online.de), Apr 24 2008

a(n)=2*A001353(n). [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Oct 26 2009]

spec := [S, {S=Sequence(Prod(Union(Z, Z), Sequence(Z), Sequence(Z)))}, unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

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} ]

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

(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)

Cf. A007913, A003699.

Sequence in context: A010749 A127865 A077839 this_sequence A162551 A073663 A155116

Adjacent sequences: A052527 A052528 A052529 this_sequence A052531 A052532 A052533

easy,nonn,nice

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jun 06 2000

Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 11 2006

Max Alekseyev (maxale(AT)gmail.com) changed a(0) to 0 and revised the entry accordingly, Nov 15 2007

Signs in definition corrected by John W. Layman, Nov 20 2007