A092184 - OEIS

%I #97 Sep 08 2022 08:45:13

%S 0,1,6,25,96,361,1350,5041,18816,70225,262086,978121,3650400,13623481,

%T 50843526,189750625,708158976,2642885281,9863382150,36810643321,

%U 137379191136,512706121225,1913445293766,7141075053841,26650854921600,99462344632561,371198523608646

%N Sequence S_6 of the S_r family.

%C 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 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

%C 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

%C a(n) is the block size of the (n-1)-th design in a sequence of multi-set designs with 2 blocks, see A335649. - _John P. McSorley_, Jun 22 2020

%H Hugo Pfoertner, <a href="/A092184/b092184.txt">Table of n, a(n) for n = 0..250</a>

%H Marco Abrate, Stefano Barbero, Umberto Cerruti, Nadir Murru, <a href="https://www.emis.de/journals/INTEGERS/papers/p38/p38.Abstract.html">Polynomial sequences on quadratic curves</a>, Integers, Vol. 15, 2015, #A38.

%H Peter Bala, <a href="/A100047/a100047.pdf">Linear divisibility sequences and Chebyshev polynomials</a>

%H Ioana-Claudia Lazăr, <a href="https://arxiv.org/abs/1904.06555">Lucas sequences in t-uniform simplicial complexes</a>, arXiv:1904.06555 [math.GR], 2019.

%H R. Stephan, <a href="http://www.ark.in-berlin.de/A001110.ps">Boring proof of a nonlinearity</a>

%H H. C. Williams and R. K. Guy, <a href="http://dx.doi.org/10.1142/S1793042111004587">Some fourth-order linear divisibility sequences</a>, Intl. J. Number Theory 7 (5) (2011) 1255-1277.

%H H. C. Williams and R. K. Guy, <a href="http://www.emis.de/journals/INTEGERS/papers/a17self/a17self.Abstract.html">Some Monoapparitic Fourth Order Linear Divisibility Sequences</a> Integers, Volume 12A (2012) The John Selfridge Memorial Volume.

%H <a href="/index/Ch#Cheby">Index entries for sequences related to Chebyshev polynomials.</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5,1).

%F 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 a(n) = 1/2*(-2 + (2+sqrt(3))^n + (2-sqrt(3))^n). - _Ralf Stephan_, Apr 14 2004

%F 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]

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

%F 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 An integer k is in this sequence iff k is nonnegative and (k^2 + 2*k)/3 is a square. - _James R. Buddenhagen_, Nov 16 2005

%F 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

%F a(n) = 4*a(n-1) - a(n-2) + 2. - _Zerinvary Lajos_, Mar 09 2008

%F From _Peter Bala_, Mar 25 2014: (Start)

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

%F 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.

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

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

%F 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.

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

%F 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

%F E.g.f.: exp(2*x)*cosh(sqrt(3)*x) - cosh(x) - sinh(x). - _Stefano Spezia_, Oct 13 2019

%e 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 := proc(n)

%p     option remember;

%p     if n <= 1 then

%p         n;

%p     else

%p         4*procname(n-1)-procname(n-2)+2 ;

%p     end if ;

%p end proc:

%p seq(A092184(n),n=0..10) ;# _Zerinvary Lajos_, Mar 09 2008

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

%t LinearRecurrence[{5,-5,1},{0,1,6},27] (* _Ray Chandler_, Jan 27 2014 *)

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

%o (PARI) Vec(x*(1+x)/(1 - 5*x + 5*x^2 - x^3) + O(x^50)) \\ _Michel Marcus_, Oct 14 2015

%o (Magma) [Floor(1/2*(-2+(2+Sqrt(3))^n+(2-Sqrt(3))^n)): n in [0..30]]; // _Vincenzo Librandi_, Oct 14 2015

%Y See A001110=S_36 for further references to S_r sequences.

%Y 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.

%K easy,nonn

%O 0,3

%A _Rainer Rosenthal_, Apr 03 2004

%E Extension and Chebyshev comments from _Wolfdieter Lang_, Sep 10 2004