

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
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OFFSET

0,3


COMMENTS

The rfamily of sequences is S_r(n) = 2*(T(n,(r2)/2)  1)/(r4) 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(n1)*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 4thorder 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 fourthorder linear divisibility sequences, Intl. J. Number Theory 7 (5) (2011) 12551277.
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(n1)*a(n+1) = (a(n)1)^2. This sequence emanates from r=6.
a(n) = 1/2*(2 + (2+sqrt(3))^n + (2sqrt(3))^n).  Ralf Stephan, Apr 14 2004
G.f.: x*(1+x)/(1  5*x + 5*x^2  x^3) = x*(1+x)/((1x)*(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(n1), 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(n1)  a(n2) + 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(n1)  u(n2) for n >= 2.
Equivalently, a(n) = U(n1,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 4thorder 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 = (61)^2 = (a(2)1)^2.


MAPLE

a[0]:=0:a[1]:=1:for n from 2 to 50 do a[n]:=4*a[n1]a[n2]+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+(2Sqrt(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 rfamily 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



