A246643 A sequence used in the touching circle problem described in A247512.

1, 10, 121, 1690, 25921, 420250, 7027801, 119508490, 2050368961, 35341836010, 610665665401, 10564982353210, 182902930753921, 3167536046903290, 54865571909148121, 950426408617182250, 16464857882672822401, 285238628280432626890, 4941562979309619843961

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