A097080 a(n) = 2*n^2 - 2*n + 3.

3, 7, 15, 27, 43, 63, 87, 115, 147, 183, 223, 267, 315, 367, 423, 483, 547, 615, 687, 763, 843, 927, 1015, 1107, 1203, 1303, 1407, 1515, 1627, 1743, 1863, 1987, 2115, 2247, 2383, 2523, 2667, 2815, 2967, 3123, 3283, 3447, 3615, 3787, 3963, 4143, 4327, 4515, 4707

Offset: 1

Author

N. J. A. Sloane, Nov 02 2008

Keywords

• nonn
• easy

Comments

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).

This ordering of the rationals is given in A113136/A113137.

The old entry with this sequence number was a duplicate of A027356.

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

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

Numbers k such that 2*k - 5 is a square. - Bruno Berselli, Nov 08 2017

References

• M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996, p. 7.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Steven Edwards and William Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., Vol. 55, No. 4 (2017), pp. 356-366.
• Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.
• Kival Ngaokrajang, Illustration of the Pappus chain of the symmetric Arbelos.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A001845, A002246, A027356, A027575, A046092, A059100, A113136, A113137.

Programs

Haskell

a097080 n = 2 * n * (n - 1) + 3  -- Reinhard Zumkeller, Dec 15 2013

Mathematica

Table[2n^2-2n+3,{n,50}] (* or *) LinearRecurrence[{3,-3,1},{3,7,15},50] (* Harvey P. Dale, Aug 02 2014 *)
CoefficientList[Series[(3 - 2 x + 3 x^2)/(1 - x)^3, {x, 0, 50}], x] (* Vincenzo Librandi, Aug 03 2014 *)

PARI

a(n)=2*n^2-2*n+3 \\ Charles R Greathouse IV, Jun 13 2012

PARI

Vec(x*(3-2*x+3*x^2)/(1-x)^3 + O(x^50)) \\ Altug Alkan, Nov 11 2015

Formula

a(n) = 4*(n-1) + a(n-1) for n > 1, a(1)=3. - Vincenzo Librandi, Nov 16 2010

a(n) = A046092(n) + 3. - Reinhard Zumkeller, Dec 15 2013

G.f.: x*(3 - 2*x + 3*x^2)/(1 - x)^3. - Vincenzo Librandi, Aug 03 2014

a(n) = A027575(n-2)/2. - Michel Marcus, Nov 11 2015

Sum_{n>=1} 1/a(n) = Pi*tanh(sqrt(5)*Pi/2)/(2*sqrt(5)). - Amiram Eldar, Dec 23 2022

From Elmo R. Oliveira, Nov 16 2024: (Start)

E.g.f.: exp(x)*(2*x^2 + 3) - 3.

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