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COMMENTS
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In general, denominators, a(k,n) and numerators, b(k,n), of continued fraction convergents to sqrt((k+1)/k) may be found as follows:
a(k,0) = 1, a(k,1) = 2k; for n > 0, a(k,2n) = 2*a(k,2n-1) + a(k,2n-2)
and a(k,2n+1) = (2k)*a(k,2n) + a(k,2n-1);
b(k,0) = 1, b(k,1) = 2k+1; for n > 0, b(k,2n) = 2*b(k,2n-1) + b(k,2n-2)
and b(k,2n+1) = (2k)*b(k,2n) + b(k,2n-1).
For example, the convergents to sqrt(4/3) start 1/1, 9/8, 19/17, 161/144, 341/305.
In general, if a(k,n) and b(k,n) are the denominators and numerators, respectively, of continued fraction convergents to sqrt((k+1)/k) as defined above, then
k*a(k,2n)^2 - a(k,2n-1)*a(k,2n+1) = k = k*a(k,2n-2)*a(k,2n) - a(k,2n-1)^2 and
b(k,2n-1)*b(k,2n+1) - k*b(k,2n)^2 = k+1 = b(k,2n-1)^2 - k*b(k,2n-2)*b(k,2n);
for example, if k=4 and n=3, then b(4,n)=a(n) and
4*a(4,6)^2 - a(4,5)*a(4,7) = 4*5473^2 - 2584*46368 = 4;
4*a(4,4)*a(4,6) - a(4,5)^2 = 4*305*5473 - 2584^2 = 4;
b(4,5)*b(4,7) - 4*b(4,6)^2 = 2889*51841 - 4*6119^2 = 5;
b(4,5)^2 - 4*b(4,4)*b(4,6) = 2889^2 - 4*341*6119 = 5.
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