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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; internal format)
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OFFSET
| 0,2
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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 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).
{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 R. H. Hardin (rhhardin(AT)att.net), 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)). [From Milan R. Janjic (agnus(AT)blic.net), 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); ex: 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), Aug 07 2010]
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REFERENCES
| Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
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LINKS
| R. Zumkeller, Enumerations of Divisors [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 17 2009]
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
| 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)=Cosh[2*ArcCosh[n] {n=0,1,2,3,...} [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
a(n)=4*n+a(n-1)-2 (with a(0)=1) [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 07 2010]
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EXAMPLE
| a(1)=3 since (0 0 / 0 0), (1 0 / 0 1) and (0 1 / 1 0) have different determinants.
a(1)=4*1+1-2=3; a(2)=4*2+3-2=9; a(3)=4*3+9-2=19 [From Vincenzo Librandi (vincenzo.librandi(AT)tin.it), Aug 07 2010]
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MAPLE
| with (combinat):seq(fibonacci(3, n)+n^2, n=0..45); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), May 25 2008
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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}]
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]
Table[Round[N[Cosh[2 ArcSinh[n]], 100]], {n, 0, 100}] (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
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PROG
| (PARI) a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011
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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 [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]
A001079, A037270, A071253, A108741, A132592, A146311, A146312, A146313, A173115,A173116 A173121. [From Artur Jasinski (grafix(AT)csl.pl), Feb 10 2010]
Column 2 of array A188645.
Sequence in context: A194139 A194115 A066506 * A049749 A147055 A146638
Adjacent sequences: A058328 A058329 A058330 * A058332 A058333 A058334
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KEYWORD
| nonn,easy
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AUTHOR
| Erich Friedman (efriedma(AT)stetson.edu), Dec 12 2000
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EXTENSIONS
| Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001
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