A047235 Numbers that are congruent to {2, 4} mod 6.

2, 4, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 40, 44, 46, 50, 52, 56, 58, 62, 64, 68, 70, 74, 76, 80, 82, 86, 88, 92, 94, 98, 100, 104, 106, 110, 112, 116, 118, 122, 124, 128, 130, 134, 136, 140, 142, 146, 148, 152, 154, 158, 160, 164, 166, 170, 172, 176, 178, 182, 184, 188, 190, 194, 196, 200, 202, 206

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( 19 ).

Complement of A047273; A093719(a(n)) = 0. - Reinhard Zumkeller, Oct 01 2008

One could prefix an initial term "1" (or not) and define this sequence through a(n+1) = a(n) + (a(n) mod 6). See A001651 for the analog with 3, A235700 (with 5), A047350 (with 7), A007612 (with 9) and A102039 (with 10). Using 4 or 8 yields a constant sequence from that term on. - M. F. Hasler, Jan 14 2014

Nonnegative m such that m^2/6 + 1/3 is an integer. - Bruno Berselli, Apr 13 2017

Numbers divisible by 2 but not by 3. - David James Sycamore, Apr 04 2018

Numbers k for which A276086(k) is of the form 6m+3. - Antti Karttunen, Dec 03 2022

Links

• David A. Corneth, Table of n, a(n) for n = 1..10000
• Chunhui Lai, A note on potentially K_4-e graphical sequences, arXiv:math/0308105 [math.CO], 2003.
• William A. Stein, The modular forms database.
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Cf. A020760, A020832, A093719, A047273 (complement), A120325 (characteristic function).

Equals 2*A001651.

Cf. A007310 ((6*n+(-1)^n-3)/2). - Bruno Berselli, Jun 24 2010

Positions of 3's in A053669 and in A358840.

Programs

Magma

[ n eq 1 select 2 else Self(n-1)+2*(1+n mod 2): n in [1..70] ]; // Klaus Brockhaus, Dec 13 2008

Maple

seq(6*floor((n+1)/2) + 3 + (-1)^n, n=1..67); # Gary Detlefs, Mar 02 2010

Mathematica

Flatten[Table[{6n - 4, 6n - 2}, {n, 40}]] (* Alonso del Arte, Oct 27 2014 *)

PARI

a(n)=(n-1)\2*6+3+(-1)^n \\ Charles R Greathouse IV, Jul 01 2013

PARI

first(n) = my(v = vector(n, i, 3*i - 1)); forstep(i = 2, n, 2, v[i]--); v \\ David A. Corneth, Oct 20 2017

Formula

a(n) = 2*A001651(n).

n such that phi(3*n) = phi(2*n). - Benoit Cloitre, Aug 06 2003

G.f.: 2*x*(1 + x + x^2)/((1 + x)*(1 - x)^2). a(n) = 3*n - 3/2 - (-1)^n/2. - R. J. Mathar, Nov 22 2008

a(n) = 3*n + 5..n odd, 3*n + 4..n even a(n) = 6*floor((n+1)/2) + 3 + (-1)^n. - Gary Detlefs, Mar 02 2010

a(n) = 6*n - a(n-1) - 6 (with a(1) = 2). - Vincenzo Librandi, Aug 05 2010

a(n+1) = a(n) + (a(n) mod 6). - M. F. Hasler, Jan 14 2014

Sum_{n>=1} 1/a(n)^2 = Pi^2/27. - Dimitris Valianatos, Oct 10 2017

a(n) = (6*n - (-1)^n - 3)/2. - Ammar Khatab, Aug 23 2020

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(6*sqrt(3)). - Amiram Eldar, Dec 11 2021

E.g.f.: 2 + ((6*x - 3)*exp(x) - exp(-x))/2. - David Lovler, Aug 25 2022

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = 2/sqrt(3) (10 * A020832).

Product_{n>=1} (1 + (-1)^n/a(n)) = 1/sqrt(3) (A020760). (End)