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

1,1

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).

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)

Mathematica

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

Crossrefs

Fifth row of A092260.

Cf. A010527, A120683.

Sequence in context: A099389 A099382 A163633 * A003630 A122565 A369105

Adjacent sequences: A092239 A092240 A092241 * A092243 A092244 A092245

Keyword

nonn,easy

Author

Giovanni Teofilatto, Feb 19 2004

Extensions

Edited and extended by Ray Chandler, Feb 21 2004

Status

approved