%I
%S 3,7,15,27,43,63,87,115,147,183,223,267,315,367,423,483,547,615,687,
%T 763,843,927,1015,1107,1203,1303,1407,1515,1627,1743,1863,1987,2115,
%U 2247,2383,2523,2667,2815,2967,3123,3283,3447,3615,3787,3963,4143,4327,4515,4707
%N a(n) = 2*n^2  2*n + 3.
%C The rational numbers may be totally ordered, first by height (see A002246) and then by magnitude; every rational number of height n appears in this ordering at a position <= a(n).
%C This ordering of the rationals is given in A113136/A113137.
%C The old entry with this sequence number was a duplicate of A027356.
%C This is also the sum of the pairwise averages of five consecutive triangular numbers in A000217. Start with A000217(0): (0+1)/2 + (1+3)/2 + (3+6)/2 + (6+10)/2 = 15, which is the third term of this sequence. Simply continue to create this sequence.  _J. M. Bergot_, Jun 13 2012
%C 2*a(n) is inverse radius (curvature) of the touching circle of the large semicircle (radius 1) and the nth and (n1)st circles of the Pappus chain of the symmetric Arbelos. One can use Descartes three circle theorem to find the solution 4*n^2  4*n + 6, n >= 1. Note that the circle with curvature 6 is also the third circle of the clockwise Pappus chain, which is A059100(2) (by symmetry). See the illustration.  _Wolfdieter Lang_ and _Kival Ngaokrajang_, Jul 01 2015
%C Numbers k such that 2*k  5 is a square.  _Bruno Berselli_, Nov 08 2017
%D Steven Edwards and W. Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., 55 (2017), 356366.
%D M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 7.
%H Vincenzo Librandi, <a href="/A097080/b097080.txt">Table of n, a(n) for n = 1..1000</a>
%H Kival Ngaokrajang, <a href="/A097080/a097080.pdf">Illustration of the Pappus chain of the symmetric Arbelos</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*(n1) + a(n1) for n>1, a(1)=3.  _Vincenzo Librandi_, Nov 16 2010
%F a(n) = A046092(n) + 3.  _Reinhard Zumkeller_, Dec 15 2013
%F G.f.: x*(3  2*x + 3*x^2)/(1  x)^3.  _Vincenzo Librandi_, Aug 03 2014
%F a(n) = A027575(n2)/2.  _Michel Marcus_, Nov 11 2015
%t Table[2n^22n+3,{n,50}] (* or *) LinearRecurrence[{3,3,1},{3,7,15},50] (* _Harvey P. Dale_, Aug 02 2014 *)
%t CoefficientList[Series[(3  2 x + 3 x^2)/(1  x)^3, {x, 0, 50}], x] (* _Vincenzo Librandi_, Aug 03 2014 *)
%o (PARI) a(n)=2*n^22*n+3 \\ _Charles R Greathouse IV_, Jun 13 2012
%o (PARI) Vec(x*(32*x+3*x^2)/(1x)^3 + O(x^50)) \\ _Altug Alkan_, Nov 11 2015
%o (Haskell)
%o a097080 n = 2 * n * (n  1) + 3  _Reinhard Zumkeller_, Dec 15 2013
%Y Cf. A001845, A002246, A059100, A113136, A113137.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_, Nov 02 2008
