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A092184 Sequence S_6 of the S_r family. 46
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; text; internal format)
OFFSET

0,3

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 R. Buddenhagen, Nov 16 2005

Taking the square root of the expression given by Buddenhagen below as necessary and sufficient for inclusion in a(n), yields A001353(n), which has a property similar to that of S_r family (see Formulas), but modified to: a(n-1)*a(n+1) = a(n)^2 - 1. - Richard R. Forberg, Sep 04 2013

This sequence is a divisibility sequence, i.e., a(n) divides a(m) whenever n divides m. It is the case P1 = 6, P2 = 8, Q = 1 of the 3 parameter family of 4th-order linear divisibility sequences found by Williams and Guy. - Peter Bala, Mar 25 2014

LINKS

Hugo Pfoertner, Table of n, a(n) for n = 0..250

Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, Polynomial sequences on quadratic curves, Integers, Vol. 15, 2015, #A38.

Peter Bala, Linear divisibility sequences and Chebyshev polynomials

R. Stephan, Boring proof of a nonlinearity

H. C. Williams and R. K. Guy, Some fourth-order linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

H. C. Williams and R. K. Guy, Some Monoapparitic Fourth Order Linear Divisibility Sequences Integers, Volume 12A (2012) The John Selfridge Memorial Volume

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (5,-5,1).

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). - Ralf 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 Ralf 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 R. Buddenhagen, 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(n) = 4*a(n-1) - a(n-2) + 2. - Richard R. Forberg, Sep 04 2013; corrected by Antonio Laface, Sep 18 2016

From Peter Bala, Mar 25 2014: (Start)

a(2*n) = 6*A001353(n)^2; a(2*n+1) = A001834(n)^2.

a(n) = u(n)^2, where {u(n)} is the Lucas sequence in the quadratic integer ring Z[sqrt(6)] defined by the recurrence u(0) = 0, u(1) = 1, u(n) = sqrt(6)*u(n-1) - u(n-2) for n >= 2.

Equivalently, a(n) = U(n-1,sqrt(6)/2)^2, where U(n,x) denotes the Chebyshev polynomial of the second kind.

a(n) = 1/2*( ((sqrt(6) + sqrt(2))/2)^n - ((sqrt(6) - sqrt(2))/2)^n )^2.

a(n) = bottom left entry of the 2 X 2 matrix T(n, M), where M is the 2 X 2 matrix [0, -2; 1, 3] and T(n,x) denotes the Chebyshev polynomial of the first kind. Cf. A098306.

See the remarks in A100047 for the general connection between Chebyshev polynomials of the first kind and 4th-order linear divisibility sequences. (End)

exp( Sum_{n >= 1} 2*a(n)*x^n/n ) = 1 + Sum_{n >= 1} A052530(n)*x^n. Cf. A001350. - Peter Bala, Mar 19 2015

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.

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, Mar 09 2008

MATHEMATICA

Table[Simplify[ -((2 + Sqrt[3])^n - 1)*((2 - Sqrt[3])^n - 1)]/2, {n, 0, 26}] (* Stefan Steinerberger, May 15 2007 *)

LinearRecurrence[{5, -5, 1}, {0, 1, 6}, 27] (* Ray Chandler, Jan 27 2014 *)

CoefficientList[Series[x (1 + x)/(1 - 5 x + 5 x^2 - x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Oct 14 2015 *)

PROG

(PARI) Vec(x*(1+x)/(1 - 5*x + 5*x^2 - x^3) + O(x^50)) \\ Michel Marcus, Oct 14 2015

(MAGMA) [Floor(1/2*(-2+(2+Sqrt(3))^n+(2-Sqrt(3))^n)): n in [0..30]]; // Vincenzo Librandi, Oct 14 2015

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). A098306, A100047. A001353, A001834. A001350, A052530.

Sequence in context: A092491 A112308 A034336 * A214955 A034559 A034347

Adjacent sequences:  A092181 A092182 A092183 * A092185 A092186 A092187

KEYWORD

easy,nonn

AUTHOR

Rainer Rosenthal, Apr 03 2004

EXTENSIONS

Extension and Chebyshev comments from Wolfdieter Lang, Sep 10 2004

STATUS

approved

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Last modified March 28 15:52 EDT 2017. Contains 284243 sequences.