%I #97 Jul 13 2024 20:28:58

%S 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,

%T 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,

%U 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131

%N 0 together with odd numbers.

%C Also continued fraction for tanh(1) (A073744 is decimal expansion). - _Rick L. Shepherd_, Aug 07 2002

%C From _Jaroslav Krizek_, May 28 2010: (Start)

%C 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.

%C 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).

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

%C 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

%C A214546(a(n)) > 0. - _Reinhard Zumkeller_, Jul 20 2012

%C Dimension of the space of weight 2n+2 cusp forms for Gamma_0(6).

%C 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

%H Vincenzo Librandi, <a href="/A004273/b004273.txt">Table of n, a(n) for n = 0..10000</a>

%H Allan Bickle, <a href="https://doi.org/10.7151/dmgt.1637">Structural results on maximal k-degenerate graphs</a>, Discuss. Math. Graph Theory 32 4 (2012), 659-676.

%H Allan Bickle, <a href="https://doi.org/10.20429/tag.2024.000105">A Survey of Maximal k-degenerate Graphs and k-Trees</a>, Theory and Applications of Graphs 0 1 (2024) Article 5.

%H D. R. Lick and A. T. White, <a href="https://doi.org/10.4153/CJM-1970-125-1">k-degenerate graphs</a>, Canad. J. Math. 22 (1970), 1082-1096.

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).

%F G.f.: x*(1+x)/(-1+x)^2. - _R. J. Mathar_, Nov 18 2007

%F a(n) = lodumo_2(A057427(n)). - _Philippe Deléham_, Apr 26 2009

%F Euler transform of length 2 sequence [3, -1]. - _Michael Somos_, Jul 03 2014

%F a(n) = (4*n - 1 - (-1)^(2^n))/2. - _Luce ETIENNE_, Jul 11 2015

%e 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 + ...

%t Join[{0}, Range[1, 200, 2]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2012 *)

%o (Magma) [2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // _Vincenzo Librandi_, Sep 21 2011

%o (Sage) def a(n) : return( dimension_cusp_forms( Gamma0(6), 2*n+2) ); # _Michael Somos_, Jul 03 2014

%o (PARI) a(n)=max(2*n-1,n) \\ _Charles R Greathouse IV_, May 14 2014

%o (GAP) Concatenation([0],List([1,3..141])); # _Muniru A Asiru_, Jul 28 2018

%o (Python)

%o def A004273(n): return (n<<1)-1 if n else 0 # _Chai Wah Wu_, Jul 13 2024

%Y 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]

%Y Cf. A005408.

%K nonn,easy

%O 0,3

%A _N. J. A. Sloane_