A004273 0 together with odd numbers.

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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 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).

a(n) = A000330(A016777(n)) / A000217(A016777(n)) = A146535(A016777(n)+1). (End)

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

Numbers not considered even for the purpose of roulette. J. Lowell, Apr 29 2025

Examples

			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 + ...
		

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).

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]

Cf. A005408.

Programs

GAP

Concatenation([0],List([1,3..141])); # Muniru A Asiru, Jul 28 2018

Magma

[2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // Vincenzo Librandi, Sep 21 2011

Mathematica

Join[{0}, Range[1, 200, 2]] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

PARI

a(n)=max(2*n-1,n) \\ Charles R Greathouse IV, May 14 2014

Python

def A004273(n): return (n<<1)-1 if n else 0 # Chai Wah Wu, Jul 13 2024

Sage

def a(n) : return( dimension_cusp_forms( Gamma0(6), 2*n+2) ); # Michael Somos, Jul 03 2014

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