%I A058331
%S A058331 1,3,9,19,33,51,73,99,129,163,201,243,289,339,393,451,513,579,649,
%T A058331 723,801,883,969,1059,1153,1251,1353,1459,1569,1683,1801,1923,2049,
%U A058331 2179,2313,2451,2593,2739,2889,3043,3201,3363,3529,3699,3873,4051
%N A058331 2*n^2 + 1.
%C A058331 Maximal number of regions in the plane that can be formed with n hyperbolas.
%C A058331 Also the number of different 2 X 2 determinants with integer entries 
               from 0 to n.
%C A058331 Number of lattice points in n-dimensional ball of radius sqrt(2). - David 
               W. Wilson (davidwwilson(AT)comcast.net), May 03 2001
%C A058331 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).
%C A058331 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]
%C A058331 {a(k): 0 <= k < 3} = divisors of 9. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 17 2009]
%C A058331 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]
%D A058331 Clark Kimberling, Complementary Equations, Journal of Integer Sequences, 
               Vol. 10 (2007), Article 07.1.4.
%H A058331 R. Zumkeller, <a href="a161700.txt">Enumerations of Divisors</a> [From 
               Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
%F A058331 G.f.: (1+3x^2)/(1-x)^3 - Paul Barry (pbarry(AT)wit.ie), Apr 06 2003
%F A058331 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
%F A058331 Equals A112295(unsigned) * [1,2,3,...]. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Oct 07 2007
%F A058331 Equals binomial transform of [1, 2, 4, 0, 0, 0,...] - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), May 03 2008
%F A058331 a(n)=4*n+a(n-1)-6 (with a(1)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Nov 09 2009]
%e A058331 a(1)=3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different 
               determinants.
%e A058331 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]
%p A058331 with (combinat):seq(fibonacci(3, n)+n^2, n=0..45); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), May 25 2008
%t A058331 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}]
%t A058331 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]
%Y A058331 Cf. A000124.
%Y A058331 Second row of array A099597.
%Y A058331 See A120062 for sequences related to integer-sided triangles with integer 
               inradius n.
%Y A058331 Cf. A112295.
%Y A058331 CF. A087113, A002552 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), 
               Feb 11 2009]
%Y A058331 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]
%Y A058331 Sequence in context: A018495 A143056 A066506 this_sequence A049749 A147055 
               A146638
%Y A058331 Adjacent sequences: A058328 A058329 A058330 this_sequence A058332 A058333 
               A058334
%K A058331 nonn,easy,new
%O A058331 0,2
%A A058331 Erich Friedman (efriedma(AT)stetson.edu), Dec 12 2000
%E A058331 Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001