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A246643 A sequence used in the touching circle problem described in A247512. 3
1, 10, 121, 1690, 25921, 420250, 7027801, 119508490, 2050368961, 35341836010, 610665665401, 10564982353210, 182902930753921, 3167536046903290, 54865571909148121, 950426408617182250, 16464857882672822401, 285238628280432626890, 4941562979309619843961 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

This sequence appears in the touching circle problem considered in A247512. There the rational curvatures are b'(n) = a(n)/9^(n-1), and A247512(n) = floor(b'(n)).

See the W. Lang link, part II) with the details where B'(n) plays the role of a(n).

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..800

Wolfdieter Lang, Curvature computation for A247335 and A247512.

Index entries for linear recurrences with constant coefficients, signature (31,-279,729).

FORMULA

One step recurrence: a(n) = 11*a(n-1) - 9^(n-1) + 20*sqrt((a(n-1) - 9^(n-1))*a(n-1)/10), a(0) = 1, n >= 1.

a(n) =  (9^n)*(1 + S(n, 22/9) - (11/9)*S(n-1, 22/9))/2, with Chebyshev's S-polynomials (see A049310). For 9^n*S(n, 22/9) see A246645(n). The positive integer sequence sqrt((a(n) - 9^n)*a(n)/10) = A246645(n-1).

O.g.f.: (1 - 21*x + 90*x^2)/((1 - 9*x)*(1 - 22*x + 81*x^2)) = (1/2)*((1 -11*x)/(1 - 22*x + 81*x^2 ) - 1/(1 - 9*x)).

For the proofs see the W. Lang link with a(n) = B'(n).

a(n) = 31*a(n-1)-279*a(n-2)+729*a(n-3). - Colin Barker, Sep 30 2014

MATHEMATICA

CoefficientList[Series[(1 - 21*x + 90*x^2)/((1 - 9*x)*(1 - 22*x + 81*x^2)), {x, 0, 50}], x] (* or *) LinearRecurrence[{31, -279, 729}, {1, 10, 121}, 50] (* G. C. Greubel, Dec 20 2017 *)

PROG

(PARI) Vec(-(6*x-1)*(15*x-1)/((9*x-1)*(81*x^2-22*x+1)) + O(x^100)) \\ Colin Barker, Sep 30 2014

(MAGMA) I:=[1, 10, 121]; [n le 3 select I[n] else 31*Self(n-1) - 279*Self(n-2) + 729*Self(n-3): n in [1..30]]; // G. C. Greubel, Dec 20 2017

CROSSREFS

Cf. A247512, A246645.

Sequence in context: A091692 A098309 A056116 * A233084 A081784 A239760

Adjacent sequences:  A246640 A246641 A246642 * A246644 A246645 A246646

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Sep 30 2014

STATUS

approved

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Last modified June 14 20:21 EDT 2021. Contains 345038 sequences. (Running on oeis4.)