

A097080


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


21



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


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


REFERENCES

Steven Edwards and W. Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., 55 (2017), 356366.
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, 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).


FORMULA

a(n) = 4*(n1) + a(n1) 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(n2)/2.  Michel Marcus, Nov 11 2015


MATHEMATICA

Table[2n^22n+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 *)


PROG

(PARI) a(n)=2*n^22*n+3 \\ Charles R Greathouse IV, Jun 13 2012
(PARI) Vec(x*(32*x+3*x^2)/(1x)^3 + O(x^50)) \\ Altug Alkan, Nov 11 2015
(Haskell)
a097080 n = 2 * n * (n  1) + 3  Reinhard Zumkeller, Dec 15 2013


CROSSREFS

Cf. A001845, A002246, A059100, A113136, A113137.
Sequence in context: A324719 A170884 A182836 * A274008 A146742 A146425
Adjacent sequences: A097077 A097078 A097079 * A097081 A097082 A097083


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane, Nov 02 2008


STATUS

approved



