A004273 - OEIS
0, 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 111, 113, 115, 117, 119, 121, 123, 125, 127, 129, 131
COMMENTS
Also continued fraction for tanh(1) (A073744 is decimal expansion). - Rick L. Shepherd, Aug 07 2002
For n >= 1, a(n) = numbers k such that arithmetic mean of the first k positive integers is an 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 an 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
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
FORMULA
G.f.: x*(1+x)/(-1+x)^2. - R. J. Mathar, Nov 18 2007
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 + ...
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
(GAP) Concatenation([0], List([1, 3..141])); # Muniru A Asiru, Jul 28 2018
(Python)
CROSSREFS
Cf. A110185, continued fraction expansion of 2*tanh(1/2), and A204877, continued fraction expansion of 3*tanh(1/3). [Bruno Berselli, Jan 26 2012]