A058331 a(n) = 2*n^2 + 1.

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

Offset: 0

Author

Erich Friedman, Dec 12 2000

Keywords

• nonn
• easy

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 A166926. - 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

Number of n-tuples from the set {1,0,-1} where at most two elements are nonzero. - Michael Somos, Oct 19 2022

a(n) gives the x-value of the integral solution (x,y) of the Pellian equation x^2 - (n^2 + 1)*y^2 = 1. The y-value is given by 2*n (see Tattersall). - Stefano Spezia, Jul 23 2025

Examples

			a(1) = 3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.
G.f. = 1 + 3*x + 9*x^2 + 19*x^3 + 33*x^4 + 51*x^5 + 73*x^6 + ... - _Michael Somos_, Oct 19 2022
		

References

• Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.
• James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 256.

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.
• Leo Tavares, Illustration: Triangular Outlines
• Reinhard Zumkeller, Enumerations of Divisors.
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

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, 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.

Cf. A000384, A000217, A166926.

Programs

Haskell

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

Magma

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

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 *)

PARI

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

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))*csch(Pi/sqrt(2)))/2. (End)

From Amiram Eldar, Feb 05 2021: (Start)

Product_{n>=0} (1 + 1/a(n)) = sqrt(2)*csch(Pi/sqrt(2))*sinh(Pi).

Product_{n>=1} (1 - 1/a(n)) = (Pi/sqrt(2))*csch(Pi/sqrt(2)). (End)

From Leo Tavares, May 23 2022: (Start)

a(n) = A000384(n+1) - 3*n.

a(n) = 3*A000217(n) + A000217(n-2). (End)

a(n) = a(-n) for all n in Z and A037235(n) = Sum_{k=0..n-1} a(k). - Michael Somos, Oct 19 2022

Extensions

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