A128385 - OEIS

%I #10 Mar 11 2014 01:32:21

%S 1,1,3,13,289,1645423,7499988983197,1716234423353399580977511919,

%T 12985299047930678223817284541389710796223289877600061663

%N a(n) = denominator of r(n): r(n) is such that the continued fraction (of rational terms) [r(1);r(2),...,r(n)] = b(n) for every positive integer n, where b(1) = 1 and b(n+1) = 1 + 1/b(n)^2 for.every positive integer n.

%C b(n) = A076725(n)/A076725(n-1)^2. The limit, as n -> infinity, of r(n)*r(n+1) = (2 /x^3) + (x^3 /2) - 2, where x is the real root of x^3 -x^2 -1 = 0. (This limit result needs some checking.)

%C a(10) has 113 digits. - _Michel Marcus_, Jan 13 2014

%e {r(n)}: 1, 1, 1/3, 9/13, 91/289,...

%e b(4) = 41/25 = 1 + 1/(1 + 1/(1/3 + 13/9)).

%e And b(5) = 2306/1681 = 1 + 1/(1 + 1/(1/3 + 1/(9/13 + 289/91))).

%o (PARI) see A128384.

%Y Cf. A128384, A076725.

%K frac,nonn

%O 1,3

%A _Leroy Quet_, Feb 28 2007

%E More terms from _Michel Marcus_, Jan 12 2014