A246639 Sequence a(n) = 3 + 5*A001519(n+1) appearing in a certain three circle touching problem, together with A246638.

8, 13, 28, 68, 173, 448, 1168, 3053, 7988, 20908, 54733, 143288, 375128, 982093, 2571148, 6731348, 17622893, 46137328, 120789088, 316229933, 827900708, 2167472188, 5674515853, 14856075368, 38893710248, 101825055373, 266581455868, 697919312228, 1827176480813, 4783610130208, 12523653909808

Offset

0,1

Comments

See the comments under A246638. The curvature c(n) defined there is c(n) = A246638(n) + (4*a(n)/5)*phi with phi = (1+sqrt(5))/2, the golden section. It lives in the quadratic number field Q(sqrt(5)). Descartes' theorem on touching circles gives c(n) = -4/5 + A(n) + A(n+1) + 2*sqrt((-4/5 )*(A(n) + A(n+1)) + A(n)*A(n+1)), with A(n) = A240926(n), n >= 0. For the proof of the first formula given below one compares this a(n) with the a(n) in c(n) given above. This uses standard Chebyshev S-polynomial identities with x = 3, like the three term recurrence and the Cassini-Simson type identity S(n, x)*S(n-2, x) = -1 + S(n-1, x)^2 (here for x=3). This implies S(n, 3)*S(n-1, 3) = (-1 + S(n, 3)^2 + S(n-1, 3)^2)/3. See also the W. Lang link in A240926, part III a).

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Index entries for sequences related to Chebyshev polynomials.

Index entries for linear recurrences with constant coefficients, signature (4,-4,1).

Formula

a(n) = 3 + 5*(S(n, 3) - S(n-1, 3)) = 3 + 5*A001519(n+1), n >= 0, with Chebyshev S-polynomials (see A049310) with S(-1, x) = 0.

O.g.f.: (8-19*x+8*x^2)/((1-x)*(1-3*x+x^2)).

a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3), n >=1, a(-2) = 13, a(-1) = 8, a(0) = 8.

a(n) = 2^(-1-n)*(3*2^(1+n)-(3-sqrt(5))^n*(-5+sqrt(5))+(3+sqrt(5))^n*(5+sqrt(5))). - Colin Barker, Nov 02 2016

Mathematica

CoefficientList[Series[(8-19*x+8*x^2)/((1-x)*(1-3*x+x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{4, -4, 1}, {8, 13, 28}, 30] (* G. C. Greubel, Dec 20 2017 *)

Prog

(PARI) Vec((8-19*x+8*x^2)/((1-x)*(1-3*x+x^2)) + O(x^30)) \\ Colin Barker, Nov 02 2016
(Magma) I:=[8, 13, 28]; [n le 3 select I[n] else 4*Self(n-1) -4*Self(n-2) + Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

Crossrefs

Cf. A246638, A049310, A001519, A115032.

Sequence in context: A337212 A093023 A057097 * A287922 A321483 A096371

Adjacent sequences: A246636 A246637 A246638 * A246640 A246641 A246642

Keyword

nonn,easy

Author

Wolfdieter Lang, Sep 01 2014

Status

approved