OFFSET
1,1
COMMENTS
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.
LINKS
Angelo B. Mingarelli, Recurrence relations and the algebraic irrationality of zeta(3), arXiv:math.NT/0608577 v1. 23 August, 2006.
FORMULA
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
a(n) ~ (1 + sqrt(2))^(8*n) / (2^(9/2) * Pi^3 * n^3). - Vaclav Kotesovec, Jul 11 2021
EXAMPLE
16*y_9/x_9 = 23.11905277493814774261896124285261449340 while
16*zeta(3)^2=23.11905277493814774261896126091180523494.
MATHEMATICA
Table[Sum[(Binomial[n, k]*Binomial[n + k, k])^2, {k, 0, n}] * Sum[(Binomial[n-1, k] Binomial[n - 1 + k, k])^2, {k, 0, n-1}], {n, 1, 20}] (* Vaclav Kotesovec, Jul 11 2021 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Angelo B. Mingarelli (amingare(AT)math.carleton.ca), Sep 10 2006
STATUS
approved