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 A063226 #62 Nov 23 2024 05:45:05
%S A063226 3,7,13,17,23,27,33,37,43,47,53,57,63,67,73,77,83,87,93,97,103,107,
%T A063226 113,117,123,127,133,137,143,147,153,157,163,167,173,177,183,187,193,
%U A063226 197,203,207,213,217,223,227,233,237,243,247
%N A063226 Dimension of the space of weight 2n cuspidal newforms for Gamma_0(63).
%C A063226 Also, dimension of the space of weight 2n cuspidal newforms for Gamma_0(88). - _N. J. A. Sloane_, Nov 24 2016
%C A063226 First differences are 4,6,4,6,4,6.... Also values of k such that k^(10*n) mod 10 = 8*(n mod 2)+1. - _Gary Detlefs_, Jul 04 2014
%C A063226 In other words, numbers n such that n^(2+4*k) + 1 is divisible by 10, for k >= 0. - _Altug Alkan_, Mar 30 2016
%C A063226 The rational generating function, the periodic first differences and Greubel's closed form are an immediate consequence of the structure of formula given by [Martin]. - _R. J. Mathar_, Apr 09 2016
%C A063226 A quasipolynomial of order 2 and degree 1: a(n) = 5n - 3 if n is even and 5n - 2 if n is odd. - _Charles R Greathouse IV_, Nov 03 2021
%C A063226 Numbers that are congruent to {3, 7} mod 10. - _Amiram Eldar_, Nov 23 2024
%H A063226 G. C. Greubel, <a href="/A063226/b063226.txt">Table of n, a(n) for n = 1..10000</a>
%H A063226 Greg Martin, <a href="http://dx.doi.org/10.1016/j.jnt.2004.10.009">Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N)</a>, J. Numb. Theory 112 (2005) 298-331, Theorem 1.
%H A063226 William A. Stein, <a href="http://www.wstein.org/Tables/dimskg0new.gp">Dimensions of the spaces S_k^{new}(Gamma_0(N))</a>.
%H A063226 William A. Stein, <a href="http://wstein.org/Tables/">The modular forms database</a>.
%H A063226 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).
%F A063226 a(n) = 4*floor(n/2) + 6*floor((n-1)/2) + 3. - _Gary Detlefs_, Jul 04 2014
%F A063226 G.f.: 3*x - x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2). - _R. J. Mathar_, Jul 15 2015
%F A063226 From _G. C. Greubel_, Mar 30 2016: (Start)
%F A063226 a(n) = (1/2)*(10*n - 5 - (-1)^n).
%F A063226 E.g.f.: (5*x + 3)*cosh(x) + (5*x + 2)*sinh(x). (End)
%F A063226 Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(5-2*sqrt(5))*Pi/10. - _Amiram Eldar_, Sep 26 2022
%F A063226 From _Amiram Eldar_, Nov 23 2024: (Start)
%F A063226 Product_{n>=1} (1 - (-1)^n/a(n)) = 2*sin(Pi/5) (A182007).
%F A063226 Product_{n>=1} (1 + (-1)^n/a(n)) = tan(Pi/5) (A019934). (End)
%p A063226 # see A063195
%t A063226 Table[4 Floor[n/2] + 6 Floor[(n - 1)/2] + 3, {n, 50}] (* or *)
%t A063226 Table[SeriesCoefficient[3 x - x^2 (-7 - 6 x + 3 x^2)/((1 + x) (x - 1)^2), {x, 0, n}], {n, 50}] (* _Michael De Vlieger_, Mar 30 2016 *)
%t A063226 LinearRecurrence[{1, 1, -1}, {3, 7, 13}, 100] (* _G. C. Greubel_, Mar 30 2016 *)
%o A063226 (PARI) my(x='x+O('x^99)); Vec(3*x-x^2*(-7-6*x+3*x^2)/((1+x)*(x-1)^2)) \\ _Altug Alkan_, Mar 31 2016
%o A063226 (PARI) a(n)=5*n-3+n%2 \\ _Charles R Greathouse IV_, Mar 31 2016
%Y A063226 Cf. A017305 (bisection), A017353 (bisection), A019934, A182007.
%K A063226 nonn,easy
%O A063226 1,1
%A A063226 _N. J. A. Sloane_, Jul 10 2001