Continued fractions are cool… Too cool for school. Or too cool for my school, at least. Because I never learnt about them there. Now that I have learnt about them, they’ve helped me wade a little further into the immeasurable Mare Mathematicum. Or Mare Matris Mathematicæ. I’m almost ankle-deep now, rather than just toe-deep. (I wish.)

But apart from aiding my understanding, continued fractions have always enhanced my entertainment. I can use them to find pretty but (probably) puny patterns like these:

[3,1,2] = contfrac(3/12) in base 9 = contfrac(3/11) in base 10
4,1,3 ← 4/13 in b16 = 4/19
5,1,4 ← 5/14 in b25 = 5/29
6,1,5 ← 6/15 in b36 = 6/41
7,1,6 ← 7/16 in b49 = 7/55
8,1,7 ← 8/17 in b64 = 8/71
9,1,8 ← 9/18 in b81 = 9/89
A,1,9 ← A/19 in b100 = 10/109 → 10,1,9
B,1,A ← B/1A in b121 = 11/131 → 11,1,10
C,1,B ← C/1B in b144 = 12/155 → 12,1,11

Those patterns with square numbers carry on for ever, I assume. I also assume that the similar patterns below do too, though I’m not sure if every base contains an infinite number of them. Maybe some bases don’t contain any at all. I haven’t found any in base 10 so far:

[25,2] = contfrac(2/52) in base 9 = contfrac(2/47) in base 10 = [23,2]
42,1,3 ← 4/213 in b8 = 4/139 → 34,1,3
4,1,2,3,3 ← 41/233 in b8 = 33/155
24,1,3,1,2 ← 24/1312 in b5 = 14/207 → 14,1,3,1,2
1,17,1,2,3 ← 117/123 in b14 = 217/227 → 1,21,1,2,3
320,1,2 ← 3/2012 in b5 = 3/257 → 85,1,2
254,2 ← 2/542 in b7 = 2/275 → 137,2
3A,3 ← 3/A3 in b28 = 3/283 → 94,3
3,5,A,2 ← 35/A2 in b34 = 107/342 → 3,5,10,2
12,1,5,3 ← 12/153 in b17 = 19/377 → 19,1,5,3
12,1,5,3 ← 12/153 in b17 = 19/377 → 19,1,5,3
3,1,4,1,4,1,5 ← 314/1415 in b8 = 204/781
2,1,36,3,2 ← 213/632 in b12 = 303/902 → 2,1,42,3,2
3,2,11,2,2,2 ← 321/1222 in b9 = 262/911 → 3,2,10,2,2,2
41,2,1,1,6 ← 41/2116 in b8 = 33/1102 → 33,2,1,1,6
4H,4 ← 4/H4 in b65 = 4/1109 → 277,4
249,2 ← 2/492 in b17 = 2/1311 → 655,2
6,2,1,3,J ← 62/13J in b35 = 212/1349 → 6,2,1,3,19
8,3,3,1,D ← 83/31D in b22 = 179/1487 → 8,3,3,1,13
142,1,1,6 ← 14/2116 in b9 = 13/1554 → 119,1,1,6
10,1,111,1,1,2 ← 1011/11112 in b6 = 223/1556 → 6,1,43,1,1,2
204,1,17 ← 20/4117 in b8 = 16/2127 → 132,1,15
93,1,8 ← 9/318 in b27 = 9/2222 → 246,1,8
1,3A,1,1,4,2,2 ← 13A1/1422 in b12 = 2281/2330 → 1,46,1,1,4,2,2
4340,1,3 ← 4/34013 in b5 = 4/2383 → 595,1,3
13,1,7,6 ← 13/176 in b46 = 49/2444 → 49,1,7,6
C7,1,B ← C/71B in b21 = 12/3119 → 259,1,11
35,3,2,3,1,1,2 ← 353/23112 in b6 = 141/3284 → 23,3,2,3,1,1,2
1,2,2,1,O,F ← 122/1OF in b50 = 2602/3715 → 1,2,2,1,24,15
2,1,1,5,55 ← 211/555 in b28 = 1597/4065 → 2,1,1,5,145
1P,2,H,2 ← 1P/2H2 in b47 = 72/5219 → 72,2,17,2
50,14,1,1,1,5 ← 501/41115 in b6 = 181/5447 → 30,10,1,1,1,5
5450,1,4 ← 5/45014 in b6 = 5/6274 → 1254,1,4
3103,1,2 ← 3/10312 in b9 = 3/6815 → 2271,1,2
4B,1,2,2,C ← 4B/122C in b19 = 87/7631 → 87,1,2,2,12
3G,D,2,3 ← 3G/D23 in b26 = 94/8843 → 94,13,2,3
3,1,1,A,K,6 ← 311/AK6 in b29 = 2553/8996 → 3,1,1,10,20,6
1,2[70],1,3,9 ← 12[70]/139 in b98 = 9870/9907 → 1,266,1,3,9
14,1,9,A ← 14/19A in b97 = 101/10292 → 101,1,9,10
14,1,9,A ← 14/19A in b97 = 101/10292 → 101,1,9,10
4133,1,14,2 ← 41/331142 in b5 = 21/11422 → 543,1,9,2
1,E,4,1,M,7 ← 1E4/1M7 in b100 = 11404/12207 → 1,14,4,1,22,7
LG,5,4 ← L/G54 in b28 = 21/12688 → 604,5,4