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A058331 a(n) = 2*n^2 + 1. 84
1, 3, 9, 19, 33, 51, 73, 99, 129, 163, 201, 243, 289, 339, 393, 451, 513, 579, 649, 723, 801, 883, 969, 1059, 1153, 1251, 1353, 1459, 1569, 1683, 1801, 1923, 2049, 2179, 2313, 2451, 2593, 2739, 2889, 3043, 3201, 3363, 3529, 3699, 3873, 4051 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Maximal number of regions in the plane that can be formed with n hyperbolas.

Also the number of different 2 X 2 determinants with integer entries from 0 to n.

Number of lattice points in an n-dimensional ball of radius sqrt(2). - David W. Wilson, May 03 2001

Equals A112295(unsigned) * [1, 2, 3, ...]. - Gary W. Adamson, Oct 07 2007

Binomial transform of [1, 2, 4, 0, 0, 0, ...] - Gary W. Adamson, May 03 2008

a(n) = longest side a of all integer-sided triangles with sides a <= b <= c and inradius n >= 1. Triangle has sides (2n^2 + 1, 2n^2 + 2, 4n^2 + 1).

{a(k): 0 <= k < 3} = divisors of 9. - Reinhard Zumkeller, Jun 17 2009

Number of ways to partition a 3*n X 2 grid into 3 connected equal-area regions. - R. H. Hardin, Oct 31 2009

Let A be the Hessenberg matrix of order n defined by: A[1, j] = 1, A[i, i] := 2, (i > 1), A[i, i - 1] = -1, and A[i, j] = 0 otherwise. Then, for n >= 3, a(n - 1) = coeff(charpoly(A, x), x^(n - 2)). - Milan Janjic, Jan 26 2010

Except for the first term of [A002522] and [A058331] if X = [A058331], Y = [A087113], A = [A002522], we have, for all other terms, Pell's equation: [A058331]^2 - [A002522]*[A087113]^2 = 1; (X^2 - A*Y^2 = 1); e.g., 3^2 -2*2^2 = 1; 9^2 - 5*4^2 = 1; 129^2 - 65*16^2 = 1, and so on. - Vincenzo Librandi, Aug 07 2010

Niven (1961) gives this formula as an example of a formula that does not contain all odd integers, in contrast to 2n + 1 and 2n - 1. - Alonso del Arte, Dec 05 2012

Numbers m such that 2*m-2 is a square. - Vincenzo Librandi, Apr 10 2015

REFERENCES

Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.

LINKS

G. C. Greubel, Table of n, a(n) for n = 0..5000

Steven Edwards and William Griffiths, On Generalized Delannoy Numbers, J. Int. Seq., Vol. 23 (2020), Article 20.3.6.

Milan Janjić, On Restricted Ternary Words and Insets, arXiv:1905.04465 [math.CO], 2019.

Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.

R. Zumkeller, Enumerations of Divisors

Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

FORMULA

G.f.: (1 + 3x^2)/(1 - x)^3. - Paul Barry, Apr 06 2003

a(n) = M^n * [1 1 1], leftmost term, where M = the 3 X 3 matrix [1 1 1 / 0 1 4 / 0 0 1]. a(0) = 1, a(1) = 3; a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). E.g., a(4) = 33 since M^4 *[1 1 1] = [33 17 1]. - Gary W. Adamson, Nov 11 2004

a(n) = cosh(2*arccosh(n)). - Artur Jasinski, Feb 10 2010

a(n) = 4*n + a(n-1) - 2 for n > 0, a(0) = 1. - Vincenzo Librandi, Aug 07 2010

a(n) = (((n-1)^2 + n^2))/2 + (n^2 + (n+1)^2)/2. - J. M. Bergot, May 31 2012

a(n) = A251599(3*n) for n > 0. - Reinhard Zumkeller, Dec 13 2014

a(n) = sqrt(8*(A000217(n-1)^2 + A000217(n)^2) + 1). - J. M. Bergot, Sep 03 2015

E.g.f.: (2*x^2 + 2*x + 1)*exp(x). - G. C. Greubel, Jul 14 2017

a(n) = A002378(n) + A002061(n). - Bruce J. Nicholson, Aug 06 2017

From Amiram Eldar, Jul 15 2020: (Start)

Sum_{n>=0} 1/a(n) = (1 + (Pi/sqrt(2))*coth(Pi/sqrt(2)))/2.

Sum_{n>=0} (-1)^n/a(n) = (1 + (Pi/sqrt(2))*cosech(Pi/sqrt(2)))/2. (End)

EXAMPLE

a(1) = 3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.

MATHEMATICA

b[g_] := Length[Union[Map[Det, Flatten[ Table[{{i, j}, {k, l}}, {i, 0, g}, {j, 0, g}, {k, 0, g}, {l, 0, g}], 3]]]] Table[b[g], {g, 0, 20}]

2*Range[0, 49]^2 + 1 (* Alonso del Arte, Dec 05 2012 *)

PROG

(PARI) a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011

(Haskell)

a058331 = (+ 1) . a001105  -- Reinhard Zumkeller, Dec 13 2014

(MAGMA) [2*n^2 + 1 : n in [0..100]]; // Wesley Ivan Hurt, Feb 02 2017

CROSSREFS

Cf. A000124.

Second row of array A099597.

See A120062 for sequences related to integer-sided triangles with integer inradius n.

Cf. A112295.

Cf. A087113, A002552.

Cf. A005408, A000124, A016813, A086514, A000125, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261.

Cf. A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115, A173116, A173121.

Column 2 of array A188645.

Cf. A001105 and A247375. - Bruno Berselli, Sep 16 2014

Cf. A056106, A251599.

Sequence in context: A194115 A226184 A066506 * A328950 A049749 A147055

Adjacent sequences:  A058328 A058329 A058330 * A058332 A058333 A058334

KEYWORD

nonn,easy

AUTHOR

Erich Friedman, Dec 12 2000

EXTENSIONS

Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001

STATUS

approved

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Last modified November 30 12:24 EST 2020. Contains 338802 sequences. (Running on oeis4.)