|
%I #20 Sep 08 2022 08:46:09
%S 1,10,121,1690,25921,420250,7027801,119508490,2050368961,35341836010,
%T 610665665401,10564982353210,182902930753921,3167536046903290,
%U 54865571909148121,950426408617182250,16464857882672822401,285238628280432626890,4941562979309619843961
%N A sequence used in the touching circle problem described in A247512.
%C 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)).
%C See the W. Lang link, part II) with the details where B'(n) plays the role of a(n).
%H G. C. Greubel, <a href="/A246643/b246643.txt">Table of n, a(n) for n = 0..800</a>
%H Wolfdieter Lang, <a href="/A246643/a246643.pdf">Curvature computation for A247335 and A247512.</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (31,-279,729).
%F 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.
%F 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).
%F 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)).
%F For the proofs see the W. Lang link with a(n) = B'(n).
%F a(n) = 31*a(n-1)-279*a(n-2)+729*a(n-3). - _Colin Barker_, Sep 30 2014
%t 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 *)
%o (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
%o (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
%Y Cf. A247512, A246645.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_, Sep 30 2014
|
