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%I #85 Jan 05 2025 19:51:37
%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 n-th and (n-1)-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 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 Steven Edwards and William Griffiths, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/55-4/EdwardsGriffiths82617.pdf">Generalizations of Delannoy and cross polytope numbers</a>, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
%H Steven Edwards and William Griffiths, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Griffiths/griffiths51.html">On Generalized Delannoy Numbers</a>, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
%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*(n-1) + a(n-1) 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(n-2)/2. - _Michel Marcus_, Nov 11 2015
%F Sum_{n>=1} 1/a(n) = Pi*tanh(sqrt(5)*Pi/2)/(2*sqrt(5)). - _Amiram Eldar_, Dec 23 2022
%F From _Elmo R. Oliveira_, Nov 16 2024: (Start)
%F E.g.f.: exp(x)*(2*x^2 + 3) - 3.
%F a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) for n > 3. (End)
%t Table[2n^2-2n+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^2-2*n+3 \\ _Charles R Greathouse IV_, Jun 13 2012
%o (PARI) Vec(x*(3-2*x+3*x^2)/(1-x)^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. A000217, A001845, A002246, A027356, A027575, A046092, A059100, A113136, A113137.
%K nonn,easy,changed
%O 1,1
%A _N. J. A. Sloane_, Nov 02 2008