0,2

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 n-dimensional ball of radius sqrt(2). - David W. Wilson (davidwwilson(AT)comcast.net), May 03 2001

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).

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); example: 3^2-2*2^2=1; 9^2-5*4^2=1, 129^2-65*16^2=1, and so on. [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

{a(k): 0 <= k < 3} = divisors of 9. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Number of ways to partition a 3*n X 2 grid into 3 connected equal-area regions. [From Ron Hardin (rhhardin(AT)att.net), Oct 31 2009]

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

R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

G.f.: (1+3x^2)/(1-x)^3 - Paul Barry (pbarry(AT)wit.ie), 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 (qntmpkt(AT)yahoo.com), Nov 11 2004

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

Equals binomial transform of [1, 2, 4, 0, 0, 0,...] - Gary W. Adamson (qntmpkt(AT)yahoo.com), May 03 2008

a(n)=4*n+a(n-1)-6 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]

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

For n=2, a(2)=4*2+1-6=3; n=2, a(2)=4*3+3-6=9; n=4, a(4)=4*4+9-6=19 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Nov 09 2009]

with (combinat):seq(fibonacci(3, n)+n^2, n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 25 2008

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}]

f[n_]:=n+(n+1)*(n+2)+(n+3)*(n+4)+(n+5); lst={}; Do[AppendTo[lst, f[n]], {n, -3, 5!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Oct 08 2009]

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 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Feb 11 2009]

A005408, A000124, A016813, A086514, A000125, A002522, A161701, A161702, A161703, A000127, A161704, A161706, A161707, A161708, A161710, A080856, A161711, A161712, A161713, A161715, A006261. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]

Sequence in context: A018495 A143056 A066506 this_sequence A049749 A147055 A146638

Adjacent sequences: A058328 A058329 A058330 this_sequence A058332 A058333 A058334

nonn,easy

Erich Friedman (efriedma(AT)stetson.edu), Dec 12 2000

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