A047233 Numbers that are congruent to {0, 4} mod 6.

0, 4, 6, 10, 12, 16, 18, 22, 24, 28, 30, 34, 36, 40, 42, 46, 48, 52, 54, 58, 60, 64, 66, 70, 72, 76, 78, 82, 84, 88, 90, 94, 96, 100, 102, 106, 108, 112, 114, 118, 120, 124, 126, 130, 132, 136, 138, 142, 144, 148, 150, 154, 156, 160, 162, 166, 168, 172, 174, 178, 180, 184, 186, 190, 192, 196, 198

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Apart from initial term(s), dimension of the space of weight 2*n cusp forms for Gamma_0(17).

Nonnegative k such that k*(k + 2)/6 is an integer. - Bruno Berselli, Mar 06 2018

Links

• Bruno Berselli, Table of n, a(n) for n = 1..10000
• William A. Stein, Dimensions of the spaces S_k(Gamma_0(N)).
• William A. Stein, The modular forms database.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A047241: (6*n - (-1)^n - 5)/2.

Cf. A342819.

Programs

Mathematica

Flatten[{#,#+4}&/@(6Range[0,30])] (* Harvey P. Dale, Jul 07 2013 *)

PARI

forstep(n=0,200,[4,2],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

Formula

From Bruno Berselli, Jun 24 2010: (Start)

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

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

a(n) = (6*n + (-1)^n - 5)/2. (End)

a(n) = 6*n - a(n-1) - 8 for n>1, a(1)=0. - Vincenzo Librandi, Aug 05 2010

a(n+1) = Sum_{k>=0} A030308(n,k)*A058764(k+1). - Philippe Deléham, Oct 17 2011

Sum_{n>=2} (-1)^n/a(n) = log(3)/4 - sqrt(3)*Pi/36. - Amiram Eldar, Dec 13 2021

E.g.f.: 2 + ((6*x -5)*exp(x) + exp(-x))/2. - David Lovler, Aug 25 2022