%I A092184
%S A092184 0,1,6,25,96,361,1350,5041,18816,70225,262086,978121,3650400,13623481,
%T A092184 50843526,189750625,708158976,2642885281,9863382150,36810643321,137379191136,
%U A092184 512706121225,1913445293766,7141075053841,26650854921600,99462344632561,
               371198523608646
%N A092184 Sequence S_6 of the S_r family.
%C A092184 The r-family of sequences is S_r(n)=2*(T(n,(r-2)/2) -1)/(r-4) provided 
               r is not equal to 4 and S_4(n)=n^2=A000290(n). Here T(n,x) are Chebyshev's 
               polynomials of the first kind. See their coefficient triangle A053120. 
               See also the R. Stephan link for the explicit formula for s_k(n) 
               for k not equal to 4 (Stephan's s_k(n) is identical with S_r(n)).
%C A092184 An integer n is in this sequence iff mutually externally tangent circles 
               with radii n, n+1, n+2 have Soddy circles (i.e. circles tangent to 
               all three) of rational radius. - James Buddenhagen (jbuddenh(AT)gmail.com), 
               Nov 16 2005
%H A092184 R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof 
               of a nonlinearity</a>
%H A092184 <a href="Sindx_Rea.html#recLCC">Index entries for sequences related to 
               linear recurrences with constant coefficients</a>
%H A092184 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A092184 S_r type sequences are defined by a(0)=0, a(1)=1, a(2)=r and a(n-1)*a(n+1) 
               = (a(n)-1)^2. This sequence emanates from r=6.
%F A092184 a(n) = 1/2*{-2+[2+sqrt(3)]^n+[2-sqrt(3)]^n}. - R. Stephan, Apr 14 2004
%F A092184 G.f.: x*(1+x)/(1-5*x+5*x^2-x^3) = x*(1+x)/((1-x)*(1-4*x+x^2)) (from the 
               R.Stephan link)
%F A092184 a(n)=T(n, 2)-1 = A001075(n)-1, with Chebyshev's polynomials T(n, 2) of 
               the first kind.
%F A092184 a(n)= b(n) + b(n-1), n>=1, with b(n):=A061278(n) the partial sums of 
               S(n, 4)= U(n, 2)= A001353(n+1) Chebyshev's polynomials of the second 
               kind.
%F A092184 An integer n is in this sequence iff n is nonnegative and (n^2 + 2*n)/
               3 is the square of an integer. - James Buddenhagen (jbuddenh(AT)gmail.com), 
               Nov 16 2005
%F A092184 a(0)=0, a(1)=1, a(n+1)=3+Floor[a(n)*(2+sqrt(3))] - Anton Vrba (antonvrba(AT)yahoo.com), 
               Jan 16 2007
%e A092184 a(3)=25 because a(1)=1 and a(2)=6 and a(1)*a(3)=1*25=(6-1)^2=(a(2)-1)^2.
%p A092184 a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n-1]-a[n-2]+2 od: seq(a[n], 
               n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 09 
               2008
%t A092184 Table[Simplify[ -((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1)]/2, {n, 
               0, 26}] - Stefan Steinerberger (stefan.steinerberger(AT)gmail.com), 
               May 15 2007.
%Y A092184 See A001110=S_36 for further references to S_r sequences.
%Y A092184 Other members of this r-family are: A007877 (r=2), |A078070| (r=3), A004146 
               (r=5), A054493 (r=7).
%Y A092184 Sequence in context: A092491 A112308 A034336 this_sequence A034559 A034347 
               A009121
%Y A092184 Adjacent sequences: A092181 A092182 A092183 this_sequence A092185 A092186 
               A092187
%K A092184 easy,nonn
%O A092184 0,3
%A A092184 Rainer Rosenthal (r.rosenthal(AT)web.de), Apr 03 2004
%E A092184 Extension and Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Sep 10 2004