|
%I #35 Oct 18 2022 07:28:14
%S 2,4,6,10,12,14,18,20,22,26,28,30,34,36,38,42,44,46,50,52,54,58,60,62,
%T 66,68,70,74,76,78,82,84,86,90,92,94,98,100,102,106,108,110,114,116,
%U 118,122,124,126,130,132,134,138,140,142,146,148,150,154,156,158
%N Numbers that are congruent to {2, 4, 6} mod 8.
%C Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 17 ).
%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 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F a(n) = 2*floor((n-1)/3) + 2*n. - _Gary Detlefs_, Mar 18 2010
%F From _R. J. Mathar_, Dec 05 2011: (Start)
%F G.f.: 2*x*(1+x)*(1+x^2) / ( (1+x+x^2)*(x-1)^2 ).
%F a(n) = 2*A042968(n). (End)
%F a(n) = a(n-1) + a(n-3) - a(n-4) for n>4, with a(1)=2, a(2)=4, a(3)=6, a(4)=10. - _Harvey P. Dale_, Oct 06 2014
%F From _Wesley Ivan Hurt_, Jun 09 2016: (Start)
%F a(n) = 2*(12*n-6-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.
%F a(3k) = 8k-2, a(3k-1) = 8k-4, a(3k-2) = 8k-6. (End)
%F Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16. - _Amiram Eldar_, Dec 19 2021
%F E.g.f.: 2*(9 + 6*exp(x)*(2*x - 1) - exp(-x/2)*(3*cos(sqrt(3)*x/2) - sqrt(3)*sin(sqrt(3)*x/2)))/9. - _Stefano Spezia_, Oct 17 2022
%p A047410:=n->2*(12*n-6-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9: seq(A047410(n), n=1..100); # _Wesley Ivan Hurt_, Jun 09 2016
%t With[{upto=140},Complement[2*Range[upto/2],8*Range[upto/8]]] (* or *) LinearRecurrence[{1,0,1,-1}, {2,4,6,10}, 60] (* _Harvey P. Dale_, Oct 06 2014 *)
%o (Magma) [n : n in [0..150] | n mod 8 in [2, 4, 6]]; // _Wesley Ivan Hurt_, Jun 09 2016
%Y Cf. A042968, A047395, A047407, A047464.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
|
