A047410 Numbers that are congruent to {2, 4, 6} mod 8.

2, 4, 6, 10, 12, 14, 18, 20, 22, 26, 28, 30, 34, 36, 38, 42, 44, 46, 50, 52, 54, 58, 60, 62, 66, 68, 70, 74, 76, 78, 82, 84, 86, 90, 92, 94, 98, 100, 102, 106, 108, 110, 114, 116, 118, 122, 124, 126, 130, 132, 134, 138, 140, 142, 146, 148, 150, 154, 156, 158

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2n cuspidal newforms for Gamma_0( 17 ).

Links

• William A. Stein, Dimensions of the spaces S_k^{new}(Gamma_0(N)).
• William A. Stein, The modular forms database.
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A042968, A047395, A047407, A047464.

Programs

Magma

[n : n in [0..150] | n mod 8 in [2, 4, 6]]; // Wesley Ivan Hurt, Jun 09 2016

Maple

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

Mathematica

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 *)

Formula

a(n) = 2*floor((n-1)/3) + 2*n. - Gary Detlefs, Mar 18 2010

From R. J. Mathar, Dec 05 2011: (Start)

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

a(n) = 2*A042968(n). (End)

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

From Wesley Ivan Hurt, Jun 09 2016: (Start)

a(n) = 2*(12*n-6-3*cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/9.

a(3k) = 8k-2, a(3k-1) = 8k-4, a(3k-2) = 8k-6. (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16. - Amiram Eldar, Dec 19 2021

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