This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A004273 #104 Apr 29 2025 21:45:58
%S A004273 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 A004273 49,51,53,55,57,59,61,63,65,67,69,71,73,75,77,79,81,83,85,87,89,91,93,
%U A004273 95,97,99,101,103,105,107,109,111,113,115,117,119,121,123,125,127,129,131
%N A004273 0 together with odd numbers.
%C A004273 Also continued fraction for tanh(1) (A073744 is decimal expansion). - _Rick L. Shepherd_, Aug 07 2002
%C A004273 From _Jaroslav Krizek_, May 28 2010: (Start)
%C A004273 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.
%C A004273 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).
%C A004273 a(n) = A000330(A016777(n)) / A000217(A016777(n)) = A146535(A016777(n)+1). (End)
%C A004273 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 A004273 A214546(a(n)) > 0. - _Reinhard Zumkeller_, Jul 20 2012
%C A004273 Dimension of the space of weight 2n+2 cusp forms for Gamma_0(6).
%C A004273 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
%C A004273 Numbers not considered even for the purpose of roulette. _J. Lowell_, Apr 29 2025
%H A004273 Vincenzo Librandi, <a href="/A004273/b004273.txt">Table of n, a(n) for n = 0..10000</a>
%H A004273 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 A004273 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 A004273 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 A004273 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2, -1).
%F A004273 G.f.: x*(1+x)/(-1+x)^2. - _R. J. Mathar_, Nov 18 2007
%F A004273 a(n) = lodumo_2(A057427(n)). - _Philippe Deléham_, Apr 26 2009
%F A004273 Euler transform of length 2 sequence [3, -1]. - _Michael Somos_, Jul 03 2014
%F A004273 a(n) = (4*n - 1 - (-1)^(2^n))/2. - _Luce ETIENNE_, Jul 11 2015
%e A004273 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 A004273 Join[{0}, Range[1, 200, 2]] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2012 *)
%o A004273 (Magma) [2*n-Floor((n+2) mod (n+1)): n in [0..70]]; // _Vincenzo Librandi_, Sep 21 2011
%o A004273 (Sage) def a(n) : return( dimension_cusp_forms( Gamma0(6), 2*n+2) ); # _Michael Somos_, Jul 03 2014
%o A004273 (PARI) a(n)=max(2*n-1,n) \\ _Charles R Greathouse IV_, May 14 2014
%o A004273 (GAP) Concatenation([0],List([1,3..141])); # _Muniru A Asiru_, Jul 28 2018
%o A004273 (Python)
%o A004273 def A004273(n): return (n<<1)-1 if n else 0 # _Chai Wah Wu_, Jul 13 2024
%Y A004273 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 A004273 Cf. A005408.
%K A004273 nonn,easy
%O A004273 0,3
%A A004273 _N. J. A. Sloane_