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 A058331 a(n) = 2*n^2 + 1. 94
 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; text; internal format)
 OFFSET 0,2 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 REFERENCES Ivan Niven, Numbers: Rational and Irrational, New York: Random House for Yale University (1961): 17. 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). 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 EXAMPLE 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 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 *) PROG (PARI) a(n)=2*n^2+1 \\ Charles R Greathouse IV, Jun 16 2011 (Haskell) a058331 = (+ 1) . a001105 -- Reinhard Zumkeller, Dec 13 2014 (Magma) [2*n^2 + 1 : n in [0..100]]; // Wesley Ivan Hurt, Feb 02 2017 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. Sequence in context: A194115 A226184 A066506 * A328950 A049749 A147055 Adjacent sequences: A058328 A058329 A058330 * A058332 A058333 A058334 KEYWORD nonn,easy AUTHOR Erich Friedman, Dec 12 2000 EXTENSIONS Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001 STATUS approved

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Last modified December 9 17:15 EST 2023. Contains 367693 sequences. (Running on oeis4.)