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
E.g.f.: (2*x^2 + 2*x + 1)*exp(x). - G. C. Greubel, Jul 14 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) = 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. 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.
KEYWORD
nonn,easy
AUTHOR
Erich Friedman, Dec 12 2000
EXTENSIONS
Revised description from Noam Katz (noamkj(AT)hotmail.com), Jan 28 2001
STATUS
approved