Search: id:A121668
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%I A121668
%S A121668 5,365,105485,47686445,27027984005,17576522979125,12539718106476125,
%T A121668 9563891779602510125,7671490770912738387125,6401115462988077760992365,
%U A121668 5513180441777884868230908125,4873728705609344219627834043125
%N A121668 Products of consecutive Apery numbers, cf. A006221.
%C A121668 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.
%D A121668 Angelo B. Mingarelli: Recurrence relations and the algebraic irrationality 
               of zeta(3), arXiv:math.NT/0608577 v1. 23 August, 2006.
%H A121668 Angelo B. Mingarelli <a href="http://arXiv.org/abs/math.NT/0608577">Recurrence 
               relations and the algebraic irrationality of zeta(3)</a>.
%F A121668 Recurrence:
%F A121668 (n + 3)^3(n + 2)^6(2n + 1)(17n^2 + 17n + 5)z(n + 2) - (2n + 1)(17n^2 
               + \
%F A121668 17n + 5)(1155n^6 + 13860n^5 + 68535n^4 + 178680n^3 + 259059n^2 + \
%F A121668 198156n + 62531)(n + 2)^3z(n + 1) + (2n + 5)(17n^2 + 85n + 107)(1155n^6 
               \
%F A121668 + 6930n^5 + 16560n^4 + 20040n^3 + 12954n^2 + 4308n + 584)(n + 1) ^3z(n) 
               \
%F A121668 - n^3(n + 1)^6(2n + 5)(17n^2 + 85n + 107)z(n - 1) = 0
%e A121668 16*y_9/x_9 = 23.11905277493814774261896124285261449340 while
%e A121668 16*zeta(3)^2=23.11905277493814774261896126091180523494.
%Y A121668 Cf. A006221.
%Y A121668 Sequence in context: A061456 A006430 A007667 this_sequence A160193 A098038 
               A072172
%Y A121668 Adjacent sequences: A121665 A121666 A121667 this_sequence A121669 A121670 
               A121671
%K A121668 easy,nonn
%O A121668 1,1
%A A121668 Angelo B. Mingarelli (amingare(AT)math.carleton.ca), Sep 10 2006

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