0,3

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)).

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

R. Stephan, Boring proof of a nonlinearity

Index entries for sequences related to linear recurrences with constant coefficients

Index entries for sequences related to Chebyshev polynomials.

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.

a(n) = 1/2*{-2+[2+sqrt(3)]^n+[2-sqrt(3)]^n}. - R. Stephan, Apr 14 2004

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)

a(n)=T(n, 2)-1 = A001075(n)-1, with Chebyshev's polynomials T(n, 2) of the first kind.

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.

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

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

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.

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

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.

See A001110=S_36 for further references to S_r sequences.

Other members of this r-family are: A007877 (r=2), |A078070| (r=3), A004146 (r=5), A054493 (r=7).

Sequence in context: A092491 A112308 A034336 this_sequence A034559 A034347 A009121

Adjacent sequences: A092181 A092182 A092183 this_sequence A092185 A092186 A092187

easy,nonn

Rainer Rosenthal (r.rosenthal(AT)web.de), Apr 03 2004

Extension and Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 10 2004