A019691 -id:A019691 - cp's OEIS frontend

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

N. J. A. Sloane

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

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).

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

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

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

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

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)

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

A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

Original entry on oeis.org

1, 0, 3, 5, 2, 7, 6, 1, 8, 0, 4, 1, 0, 0, 8, 3, 0, 4, 9, 3, 9, 5, 5, 9, 5, 3, 5, 0, 4, 9, 6, 1, 9, 3, 3, 1, 3, 3, 9, 6, 2, 7, 5, 6, 0, 5, 2, 7, 9, 7, 2, 2, 0, 5, 5, 2, 5, 6, 0, 1, 2, 8, 2, 9, 2, 6, 0, 2, 2, 7, 8, 9, 8, 9, 9, 5, 2, 0, 7, 9, 8, 7, 6, 8, 9, 4, 7, 1, 8, 9, 8, 7, 7, 6, 9, 9, 8, 6, 6, 2, 0, 8, 3, 5, 8

Offset: 1

Rick L. Shepherd, Jun 24 2006

Side length of the largest equilateral triangle that can be inscribed in a unit square (as stated in MathWorld/Weisstein link).

A quartic integer. - Charles R Greathouse IV, Aug 27 2017

			1.03527618041008304939559535049619331339627560527972...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 8.2, p. 487.

Eric Weisstein's World of Mathematics, Equilateral Triangle.
Index entries for algebraic numbers, degree 4.

Cf. A002193, A019679, A019691, A019884, A010464, A092242, A101263.

RealDigits[Sec[15 Degree],10,120][[1]] (* Harvey P. Dale, Jun 03 2015 *)

sqrt(6) - sqrt(2) \\ Charles R Greathouse IV, Aug 27 2017

Equals sec(Pi/12) = sec(A019679) = sqrt(6) - sqrt(2) = A010464 - A002193 = csc(5*Pi/12) = 1/sin(5*Pi/12) = 1/sin(10*A019691) = 1/A019884.
Equals Product_{k >= 1} 1/(1 - 1/(36*(2*k - 1)^2)). - Antonio Graciá Llorente, Mar 20 2024
From Amiram Eldar, Nov 24 2024: (Start)
Equals 2*A101263.
Equals Product_{k>=1} (1 - (-1)^k/A092242(k)). (End)
Smallest positive of the 4 real-valued roots of x^4-16*x^2+16=0. - R. J. Mathar, Aug 31 2025

A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

Original entry on oeis.org

9, 6, 0, 5, 9, 1, 9, 5, 6, 4, 5, 5, 0, 5, 2, 9, 5, 9, 4, 2, 5, 1, 0, 7, 9, 5, 1, 3, 9, 3, 8, 0, 6, 3, 6, 0, 2, 4, 0, 9, 7, 6, 9, 0, 7, 5, 4, 5, 7, 2, 3, 9, 8, 7, 6, 9, 0, 8, 9, 8, 5, 1, 5, 3, 1, 0, 3, 8, 7, 6, 6, 3, 3, 4, 0, 1, 6, 3, 2, 8, 9, 0, 3, 1, 2, 2, 7, 9, 3, 5, 6, 9, 1, 7, 7, 4, 8, 2, 4, 5, 3, 1, 2, 1, 6

Offset: 0

R. J. Mathar, Aug 10 2007

			0.960591956455052959425107951...

D. H. Bailey, J. M. Borwein and R. E. Crandall, Box Integrals, J. Comp. Appl. Math., Vol. 206, No. 1 (2007), pp. 196-208.
D. H. Bailey, J. M. Borwein, and R. E. Crandall, Advances in the theory of box integrals, Math. Comp. 79 (271) (2010) 1839-1866, Table 2. [From _R. J. Mathar_, Oct 13 2010]
Eric Weisstein's World of Mathematics, Box Integral.

Cf. A010527, A019691, A065914, A135691.

Analogous constants: A244921 (square), A254979 (4-cube).

evalf( sqrt(3)/4+log(2+sqrt(3))/2-Pi/24);

RealDigits[Sqrt[3]/4 + Log[2+Sqrt[3]]/2 - Pi/24, 10, 120][[1]] (* Amiram Eldar, Jun 04 2023 *)

Equals sqrt(3)/4 + log(2+sqrt(3))/2 - Pi/24 = A010527/2 + A065914/2 - A019691.

Equals 2 * A135691. - Amiram Eldar, Jun 04 2023

Name corrected by Amiram Eldar, Jun 04 2023

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

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A120683 Decimal expansion of secant of 15 degrees (cosecant of 75 degrees).

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A130590 Decimal expansion of the mean Euclidean distance from a point in the unit 3D cube to a given vertex of the cube.

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