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A092184
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Sequence S_6 of the S_r family.
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43
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0, 1, 6, 25, 96, 361, 1350, 5041, 18816, 70225, 262086, 978121, 3650400, 13623481, 50843526, 189750625, 708158976, 2642885281, 9863382150, 36810643321, 137379191136, 512706121225, 1913445293766, 7141075053841, 26650854921600, 99462344632561, 371198523608646
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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
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LINKS
| 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.
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FORMULA
| 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
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EXAMPLE
| 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.
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MAPLE
| 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
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MATHEMATICA
| 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.
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CROSSREFS
| 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 * A034559 A034347 A009121
Adjacent sequences: A092181 A092182 A092183 * A092185 A092186 A092187
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KEYWORD
| easy,nonn
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AUTHOR
| Rainer Rosenthal (r.rosenthal(AT)web.de), Apr 03 2004
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EXTENSIONS
| Extension and Chebyshev comments from W. Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Sep 10 2004
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