OFFSET
0,3
COMMENTS
Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
From Jaroslav Krizek, May 28 2010: (Start)
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is integer. A040001(a(n)) = 1. See A145051 and A040001.
For n >= 1, a(n) = corresponding values of antiharmonic means to numbers from A016777 (numbers k such that antiharmonic mean of the first k positive integers is integer).
If the n-th prime is denoted by p(n) then it appears that a(j) = distinct, increasing values of (Sum of the quadratic non-residues of p(n) - Sum of the quadratic residues of p(n)) / p(n) for each j. - Christopher Hunt Gribble, Oct 05 2010
A214546(a(n)) > 0. - Reinhard Zumkeller, Jul 20 2012
Dimension of the space of weight 2n+2 cusp forms for Gamma_0(6).
The size of a maximal 2-degenerate graph of order n-1 (this class includes 2-trees and maximal outerplanar graphs (MOPs)). - Allan Bickle, Nov 14 2021
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Allan Bickle, Structural results on maximal k-degenerate graphs, Discuss. Math. Graph Theory 32 4 (2012), 659-676.
Allan Bickle, A Survey of Maximal k-degenerate Graphs and k-Trees, Theory and Applications of Graphs 0 1 (2024) Article 5.
D. R. Lick and A. T. White, k-degenerate graphs, Canad. J. Math. 22 (1970), 1082-1096.
Index entries for linear recurrences with constant coefficients, signature (2, -1).
FORMULA
G.f.: x*(1+x)/(-1+x)^2. - R. J. Mathar, Nov 18 2007
a(n) = lodumo_2(A057427(n)). - Philippe Deléham, Apr 26 2009
Euler transform of length 2 sequence [3, -1]. - Michael Somos, Jul 03 2014
a(n) = (4*n - 1 - (-1)^(2^n))/2. - Luce ETIENNE, Jul 11 2015
EXAMPLE
G.f. = x + 3*x^2 + 5*x^3 + 7*x^4 + 9*x^5 + 11*x^6 + 13*x^7 + 15*x^8 + 17*x^9 + ...
MATHEMATICA
Join[{0}, Range[1, 200, 2]] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)
PROG
(Magma) [2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // Vincenzo Librandi, Sep 21 2011
(Sage) def a(n) : return( dimension_cusp_forms( Gamma0(6), 2*n+2) ); # Michael Somos, Jul 03 2014
(PARI) a(n)=max(2*n-1, n) \\ Charles R Greathouse IV, May 14 2014
(GAP) Concatenation([0], List([1, 3..141])); # Muniru A Asiru, Jul 28 2018
(Python)
def A004273(n): return (n<<1)-1 if n else 0 # Chai Wah Wu, Jul 13 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved