%I #40 Sep 11 2024 17:54:14

%S 0,1,3,3,5,5,7,7,9,9,11,11,13,13,15,15,17,17,19,19,21,21,23,23,25,25,

%T 27,27,29,29,31,31,33,33,35,35,37,37,39,39,41,41,43,43,45,45,47,47,49,

%U 49,51,51,53,53,55,55,57,57,59,59,61,61,63,63,65,65,67,67,69,69,71,71,73,73,75,75,77,77,79,79,81,81,83

%N Dimension of the space of weight 2n cuspidal newforms for Gamma_0( 7 ).

%C Also, for n>1, number of involutions (i.e. elements of order 2) in the dihedral group D_(n-1). - _Lekraj Beedassy_, Oct 22 2004

%C Also, the chromatic number of the n-th triangular graph; i.e., the chromatic index (edge chromatic number) of the n-th complete graph. - _Danny Rorabaugh_, Nov 26 2018

%H Vincenzo Librandi, <a href="/A063196/b063196.txt">Table of n, a(n) for n = 1..1000</a>

%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/WallisFormula.html">MathWorld: Wallis Formula</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>.

%H William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ChromaticNumber.html">Chromatic Number</a>, <a href="http://mathworld.wolfram.com/EdgeChromaticNumber.html">Edge Chromatic Number</a>, and <a href="http://mathworld.wolfram.com/TriangularGraph.html">Triangular Graph</a>.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

%F For n > 1, a(n-1) = (2n + 1 + (-1)^n)/2 (odd numbers appearing twice). - _Lekraj Beedassy_, Oct 22 2004

%F For n > 1, a(n) = 2*n - a(n-1), (with a(1)=1). - _Vincenzo Librandi_, Dec 06 2010

%F From _Colin Barker_, Sep 08 2013: (Start)

%F a(n) = a(n-1) + a(n-2) - a(n-3) for n > 4.

%F G.f.: -x^2*(x^2-2*x-1) / ((x-1)^2*(x+1)). (End)

%t CoefficientList[Series[-x (x^2 - 2 x - 1) / ((x - 1)^2 (x + 1)), {x, 0, 100}], x] (* _Vincenzo Librandi_, Nov 27 2018 *)

%t LinearRecurrence[{1,1,-1},{0,1,3,3},90] (* _Harvey P. Dale_, Sep 11 2024 *)

%o (PARI) concat([0], Vec(-x^2*(x^2-2*x-1)/((x-1)^2*(x+1)) + O(x^100))) \\ _Colin Barker_, Sep 08 2013

%Y Cf. A109613.

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, Jul 10 2001