A092242 Numbers that are congruent to {5, 7} (mod 12).

5, 7, 17, 19, 29, 31, 41, 43, 53, 55, 65, 67, 77, 79, 89, 91, 101, 103, 113, 115, 125, 127, 137, 139, 149, 151, 161, 163, 173, 175, 185, 187, 197, 199, 209, 211, 221, 223, 233, 235, 245, 247, 257, 259, 269, 271, 281, 283, 293, 295, 305, 307, 317, 319, 329, 331

Offset: 1

Author

Giovanni Teofilatto, Feb 19 2004

Keywords

• nonn
• easy

References

• L. B. W. Jolley, Summation of Series, Dover Publications, 1961, p. 64.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Crossrefs

Fifth row of A092260.

Cf. A010527, A120683.

Programs

Mathematica

Select[Range[331], MemberQ[{5, 7}, Mod[#, 12]] &] (* Amiram Eldar, Dec 04 2021 *)

Formula

1/5^2 + 1/7^2 + 1/17^2 + 1/19^2 + 1/29^2 + 1/31^2 + ... = Pi^2*(2 - sqrt(3))/36 = 0.073459792... [Jolley] - Gary W. Adamson, Dec 20 2006

a(n) = 12*n - a(n-1) - 12 (with a(1)=5). - Vincenzo Librandi, Nov 16 2010

From R. J. Mathar, Oct 08 2011: (Start)

a(n) = 6*n - 3 - 2*(-1)^n.

G.f.: x*(5+2*x+5*x^2) / ( (1+x)*(x-1)^2 ). (End)

Sum_{n>=1} (-1)^(n+1)/a(n) = (2 - sqrt(3))*Pi/12. - Amiram Eldar, Dec 04 2021

From Amiram Eldar, Nov 24 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/12) (A120683).

Product_{n>=1} (1 + (-1)^n/a(n)) = (sqrt(3)/2)*sec(Pi/12) (= A010527 * A120683). (End)

Extensions

Edited and extended by Ray Chandler, Feb 21 2004