A259054 - OEIS

%I #30 Nov 17 2024 12:28:34

%S 19,27,43,67,99,139,187,243,307,379,459,547,643,747,859,979,1107,1243,

%T 1387,1539,1699,1867,2043,2227,2419,2619,2827,3043,3267,3499,3739,

%U 3987,4243,4507,4779,5059,5347,5643,5947,6259,6579,6907,7243,7587,7939,8299,8667,9043,9427,9819

%N a(n) = 4*n^2 - 4*n + 19, n >= 1.

%C a(n) gives twice the inverse radius of the circles touching the large Arbelos (2/3,1/3) circle (radius 1) and the n-th and (n-1)-th circles of the counterclockwise Pappus chain.

%C For twice the curvatures (inverse radii) of the counterclockwise Pappus chain of the (2/3,1/3) arbelos see A114949, also for the MathWorld link to Pappus chain.

%C For the small curvatures touching the left circle of the (2/3,1/3) arbelos and the n-th and (n-1)-st circles of the counterclockwise Pappus chain see A259555.

%C The curvatures of the circles can be computed with Descartes' three (actually 5) circle theorem. See A259555 for links to Descartes' theorem.

%H Andrew Howroyd, <a href="/A259054/b259054.txt">Table of n, a(n) for n = 1..1000</a>

%H Kival Ngaokrajang, <a href="/A259054/a259054.pdf">Illustration of initial terms</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).

%F a(n) = 4*n^2 - 4*n + 19, n >= 1.

%F O.g.f.: x*(19 - 30*x + 19*x^2)/(1 - x)^3.

%F From _Elmo R. Oliveira_, Nov 17 2024: (Start)

%F E.g.f.: exp(x)*(4*x^2 + 19) - 19.

%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)

%t Array[4 #^2 - 4 # + 19 &, 40] (* _Michael De Vlieger_, Jul 02 2015 *)

%t LinearRecurrence[{3,-3,1},{19,27,43},40] (* _Harvey P. Dale_, May 17 2016 *)

%o (PARI) vector(60, n, 4*n^2 - 4*n + 19) \\ _Michel Marcus_, Jul 03 2015

%Y Cf. A114949, A259555.

%K nonn,easy

%O 1,1

%A _Wolfdieter Lang_ and _Kival Ngaokrajang_, Jul 02 2015

%E Terms a(37) and beyond from _Andrew Howroyd_, May 01 2020