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A246639 Sequence a(n) = 3 + 5*A001519(n+1) appearing in a certain three circle touching problem, together with A246638. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified June 25 10:28 EDT 2021. Contains 345453 sequences. (Running on oeis4.)