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146 DANIEL J. MADDEN<br />
b, 2b(2bn + 1) k−1 , b(2bn + 1), 2b(2bn + 1) k−2 , b(2bn + 1) 2 , . . . ,<br />
b(2bn + 1) k−1 , b(2bn + 1) 1 , 2b(2bn + 1) k−1 , 2b,<br />
b(2bn + 1) k + n,<br />
2b, b(2bn + 1) k−1 , 2b(2bn + 1), b(2bn + 1) k−2 , 2b(2bn + 1) 2 , . . . ,<br />
2b(2bn + 1) k−2 , b(2bn + 1) 1 , 2b(2bn + 1) k−1 , b, 2b(2bn + 1) k + 2n<br />
If we take n = b, we get<br />
<br />
b 2 ((2b 2 + 1) k + 1) 2 + 2(2b 2 + 1) k<br />
<br />
=<br />
=<br />
b2 (2b2 + 1) 2k + 2(b2 + 1)(2b2 + 1) k + 1<br />
<br />
b(2b 2 + 1) k + b;<br />
b, 2b(2b 2 + 1) k−1 , b(2b 2 + 1), 2b(2b 2 + 1) k−2 , b(2b 2 + 1) 2 , . . . ,<br />
b(2b 2 + 1) k−2 , 2b(2b 2 + 1) 1 , b(2b 2 + 1) k−1 , 2b,<br />
b(2b 2 + 1) k + b,<br />
2b, b(2b 2 + 1) k−1 , 2b(2b 2 + 1), b(2b 2 + 1) k−2 , 2b(2b 2 + 1) 2 , . . . ,<br />
2b(2b 2 + 1) k−2 , b(2b 2 + 1) 1 , 2b(2b 2 + 1) k−1 , b, 2b(2b 2 + 1) k + 2b<br />
We can multiply this by b to get a new expansion:<br />
<br />
=<br />
b 4 (2b 2 + 1) 2k + 2b 2 (b 2 + 1)(2b 2 + 1) k + b 2<br />
<br />
b 2 (2b 2 + 1) k + b 2 ;<br />
1, 2b 2 (2b 2 + 1) k−1 , (2b 2 + 1), 2b 2 (2b 2 + 1) k−2 , (2b 2 + 1) 2 , . . . ,<br />
(2b 2 + 1) k−2 , 2b 2 (2b 2 + 1) 1 , (2b 2 + 1) k−1 , 2b 2 ,<br />
(2b 2 + 1) k + 1,<br />
2b 2 , (2b 2 + 1) k−1 , 2b 2 (2b 2 + 1), (2b 2 + 1) k−2 , 2b 2 (2b 2 + 1) 2 , . . . ,<br />
2b 2 (2b 2 + 1) k−2 , (2b 2 + 1) 1 , 2b 2 (2b 2 + 1) k−1 , 1, 2b 2 (2b 2 + 1) k + 2b 2<br />
.<br />
References<br />
[A] T. Azuhatu, On the fundamental units and class numbers of real quadratic fields,<br />
II, Tokyo Journal of Math., 10 (1987), 259-270.<br />
[LR] C. Levesque and G. Rind, A few classes of periodic continued fractions, Utilitas<br />
Math, 30 (1986), 79-107.<br />
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