--- /dev/null
+! Copyright (c) 2009 Guillaume Nargeot.
+! See http://factorcode.org/license.txt for BSD license.
+USING: kernel math lists lists.lazy project-euler.common sequences ;
+IN: project-euler.065
+
+! http://projecteuler.net/index.php?section=problems&id=065
+
+! DESCRIPTION
+! -----------
+
+! The square root of 2 can be written as an infinite continued fraction.
+
+! 1
+! √2 = 1 + -------------------------
+! 1
+! 2 + ---------------------
+! 1
+! 2 + -----------------
+! 1
+! 2 + -------------
+! 2 + ...
+
+! The infinite continued fraction can be written, √2 = [1;(2)], (2) indicates
+! that 2 repeats ad infinitum. In a similar way, √23 = [4;(1,3,1,8)].
+
+! It turns out that the sequence of partial values of continued fractions for
+! square roots provide the best rational approximations. Let us consider the
+! convergents for √2.
+
+! 1 3 1 7 1 17 1 41
+! 1 + - = - ; 1 + ----- = - ; 1 + --------- = -- ; 1 + ------------- = --
+! 2 2 1 5 1 12 1 29
+! 2 + - 2 + ----- 2 + ---------
+! 2 1 1
+! 2 + - 2 + -----
+! 2 1
+! 2 + -
+! 2
+
+! Hence the sequence of the first ten convergents for √2 are:
+! 1, 3/2, 7/5, 17/12, 41/29, 99/70, 239/169, 577/408, 1393/985, 3363/2378, ...
+
+! What is most surprising is that the important mathematical constant,
+! e = [2; 1,2,1, 1,4,1, 1,6,1 , ... , 1,2k,1, ...].
+
+! The first ten terms in the sequence of convergents for e are:
+! 2, 3, 8/3, 11/4, 19/7, 87/32, 106/39, 193/71, 1264/465, 1457/536, ...
+
+! The sum of digits in the numerator of the 10th convergent is 1+4+5+7=17.
+
+! Find the sum of digits in the numerator of the 100th convergent of the
+! continued fraction for e.
+
+
+! SOLUTION
+! --------
+
+<PRIVATE
+
+: (e-frac) ( -- seq )
+ 2 lfrom [
+ dup 3 mod zero? [ 3 / 2 * ] [ drop 1 ] if
+ ] lazy-map ;
+
+: e-frac ( n -- n )
+ 1 - (e-frac) ltake list>array reverse 0
+ [ + recip ] reduce 2 + ;
+
+PRIVATE>
+
+: euler065 ( -- answer )
+ 100 e-frac numerator number>digits sum ;
+
+! [ euler065 ] 100 ave-time
+! 4 ms ave run time - 0.33 SD (100 trials)
+
+SOLUTION: euler065
project-euler.037 project-euler.038 project-euler.039 project-euler.040
project-euler.041 project-euler.042 project-euler.043 project-euler.044
project-euler.045 project-euler.046 project-euler.047 project-euler.048
- project-euler.049 project-euler.051 project-euler.052 project-euler.053
- project-euler.054 project-euler.055 project-euler.056 project-euler.057
- project-euler.058 project-euler.059 project-euler.063 project-euler.067
- project-euler.069 project-euler.071 project-euler.072 project-euler.073
- project-euler.074 project-euler.075 project-euler.076 project-euler.079
- project-euler.085 project-euler.092 project-euler.097 project-euler.099
- project-euler.100 project-euler.102 project-euler.112 project-euler.116
- project-euler.117 project-euler.124 project-euler.134 project-euler.148
- project-euler.150 project-euler.151 project-euler.164 project-euler.169
- project-euler.173 project-euler.175 project-euler.186 project-euler.190
- project-euler.203 project-euler.215 ;
+ project-euler.049 project-euler.051 project-euler.052 project-euler.053
+ project-euler.054 project-euler.055 project-euler.056 project-euler.057
+ project-euler.058 project-euler.059 project-euler.063 project-euler.065
+ project-euler.067 project-euler.069 project-euler.071 project-euler.072
+ project-euler.073 project-euler.074 project-euler.075 project-euler.076
+ project-euler.079 project-euler.085 project-euler.092 project-euler.097
+ project-euler.099 project-euler.100 project-euler.102 project-euler.112
+ project-euler.116 project-euler.117 project-euler.124 project-euler.134
+ project-euler.148 project-euler.150 project-euler.151 project-euler.164
+ project-euler.169 project-euler.173 project-euler.175 project-euler.186
+ project-euler.190 project-euler.203 project-euler.215 ;
IN: project-euler
<PRIVATE