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A097080 a(n) = 2*n^2 - 2*n + 3. 20
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 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

Steven Edwards and W. Griffiths, Generalizations of Delannoy and cross polytope numbers, Fib. Q., 55 (2017), 356-366.

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

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*(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

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 *)

PROG

(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

(Haskell)

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

CROSSREFS

Cf. A001845, A002246, A059100, A113136, A113137.

Sequence in context: A213215 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

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Last modified February 25 06:31 EST 2018. Contains 299643 sequences. (Running on oeis4.)