1,1

The solutions x_{n-1}:=A_nA_{n-1}, y_n of the four-term recurrence relation defined by x_0=5, x_1= 365, x_2= 105485 and y_0= 0, y_1=8424, y_2= 2438709 are such that y_n/x_n -> 16*zeta(3)^2. Generalizations to products of three or more Apery numbers are to be found in the cited paper.

Angelo B. Mingarelli: Recurrence relations and the algebraic irrationality of zeta(3), arXiv:math.NT/0608577 v1. 23 August, 2006.

Angelo B. Mingarelli Recurrence relations and the algebraic irrationality of zeta(3).

Recurrence:

(n + 3)^3(n + 2)^6(2n + 1)(17n^2 + 17n + 5)z(n + 2) - (2n + 1)(17n^2 + \

17n + 5)(1155n^6 + 13860n^5 + 68535n^4 + 178680n^3 + 259059n^2 + \

198156n + 62531)(n + 2)^3z(n + 1) + (2n + 5)(17n^2 + 85n + 107)(1155n^6 \

+ 6930n^5 + 16560n^4 + 20040n^3 + 12954n^2 + 4308n + 584)(n + 1) ^3z(n) \

- n^3(n + 1)^6(2n + 5)(17n^2 + 85n + 107)z(n - 1) = 0

16*y_9/x_9 = 23.11905277493814774261896124285261449340 while

16*zeta(3)^2=23.11905277493814774261896126091180523494.

Cf. A006221.

Sequence in context: A061456 A006430 A007667 this_sequence A160193 A098038 A072172

Adjacent sequences: A121665 A121666 A121667 this_sequence A121669 A121670 A121671

easy,nonn

Angelo B. Mingarelli (amingare(AT)math.carleton.ca), Sep 10 2006