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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
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OFFSET
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0,2
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COMMENTS
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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, 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. [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); ex: 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
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REFERENCES
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Clark Kimberling, Complementary Equations, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.4.
Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17.
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LINKS
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Table of n, a(n) for n=0..45.
R. Zumkeller, Enumerations of Divisors
Index entries for sequences related to linear recurrences with constant coefficients, signature (3,-3,1).
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FORMULA
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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
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) = cosh(2*arccosh(n)). [Artur Jasinski, Feb 10 2010]
a(n)=4*n + a(n - 1) - 2 (with 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
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EXAMPLE
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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 [Vincenzo Librandi, Aug 07 2010]
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MAPLE
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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
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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 (* Vladimir Joseph Stephan Orlovsky, Oct 08 2009 *)
Table[Round[N[Cosh[2 ArcSinh[n]], 100]], {n, 0, 100}] (* Artur Jasinski, Feb 10 2010 *)
2*Range[0, 49]^2 + 1 (* Alonso del Arte, Dec 05 2012 *)
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PROG
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(PARI) a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011
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CROSSREFS
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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.
Sequence in context: A194139 A194115 A066506 * A049749 A147055 A146638
Adjacent sequences: A058328 A058329 A058330 * A058332 A058333 A058334
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KEYWORD
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nonn,easy,changed
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AUTHOR
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Erich Friedman, Dec 12 2000
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EXTENSIONS
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Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001
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STATUS
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approved
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