A017593 a(n) = 12*n + 6.

6, 18, 30, 42, 54, 66, 78, 90, 102, 114, 126, 138, 150, 162, 174, 186, 198, 210, 222, 234, 246, 258, 270, 282, 294, 306, 318, 330, 342, 354, 366, 378, 390, 402, 414, 426, 438, 450, 462, 474, 486, 498, 510, 522, 534, 546, 558, 570, 582, 594, 606, 618, 630, 642

Offset: 0

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(73).

Continued fraction expansion of tanh(1/6). - Benoit Cloitre, Dec 17 2002

Also solutions to 5^x + 7^x == 11 (mod 13). - Cino Hilliard, May 10 2003

Numbers m such that the sum of the m-th powers of all 2 X 2 matrices over Z/mZ is a nonzero matrix. - José María Grau Ribas, Jan 31 2014

Positive numbers k for which 1/2 + k/4 + k^2/6 is an integer. - Bruno Berselli, Apr 12 2018

Links

• P. Fortuny, J. M. Grau, A. M. Oller-Marcén and I. F. Rúa, On power sums of matrices over a finite commutative ring, arXiv:1505.08132 [math.RA], 2015.
• Milan Janjic and Boris Petkovic, A Counting Function, arXiv 1301.4550 [math.CO], 2013.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
• 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 (2,-1).

Crossrefs

Cf. A005408, A016825, A016945, A017641, A019691, A030133.

Programs

Mathematica

12 Range[0, 200] + 6 (* Vladimir Joseph Stephan Orlovsky, Feb 19 2011 *)
LinearRecurrence[{2, -1}, {6, 18}, 60] (* Harvey P. Dale, Aug 20 2014 *)

PARI

a(n)=12*n+6 \\ Charles R Greathouse IV, Sep 24 2015

Sage

[i+6 for i in range(645) if gcd(i,12) == 12] # Zerinvary Lajos, May 21 2009

Formula

A030133(a(n)) = 9. - Reinhard Zumkeller, Jul 04 2007

a(n) = 24*n - a(n-1) with n > 0, a(0)=6. - Vincenzo Librandi, Nov 19 2010

a(0)=6, a(1)=18; for n > 1, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Aug 20 2014

G.f.: 6*(1+x)/(1-x)^2. - Wolfdieter Lang, Oct 27 2020

Sum_{n>=0} (-1)^n/a(n) = Pi/24 (A019691). - Amiram Eldar, Dec 12 2021

From Amiram Eldar, Nov 24 2024: (Start)

Product_{n>=0} (1 - (-1)^n/a(n)) = sqrt(2) * sin(5*Pi/24).

Product_{n>=0} (1 + (-1)^n/a(n)) = sqrt(2) * cos(5*Pi/24). (End)

From Elmo R. Oliveira, Apr 04 2025: (Start)

E.g.f.: 6*exp(x)*(1 + 2*x).

a(n) = 6*A005408(n) = 3*A016825(n) = 2*A016945(n). (End)

Extensions

Typos in sequence (270 was 2,70 and 510 was 5,10) fixed by Peter Luschny, Dec 14 2009