[euler] solution for problem 65

categories/euler/prob065-andreoss.pl

@@ -0,0 +1,69 @@
+#!/usr/bin/env perl6
+use v6;
+
+=begin pod
+
+=TITLE Square root digital expansion
+
+=AUTHOR Andrei Osipov
+
+L<https://projecteuler.net/problem=65>
+
+
+
+The square root of 2 can be written as an infinite continued fraction.
+
+√2 = 1 +  1
+        ______
+	2 +  1
+           ______
+  	   2 + 	1
+  	      ______
+	      2 +  1
+  	  	 ______
+	         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 + 1
+   ___              = 3/2
+    2 
+
+1 +    1
+    _________       = 7/5
+    2 +  1 / 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.
+
+
+Expected result:  272
+
+=end pod		      
+
+sub continued-fraction(@sequence, :$depth)  {
+    my $x = @sequence.shift;
+    return 1 if $depth == 1;
+    $x + 1.FatRat /
+	continued-fraction :depth($depth - 1), @sequence
+}
+
+my @e := gather { take 2;  take (1; $_; 1) for 2,4 ... * };
+
+say [+] continued-fraction(@e, depth => 100).numerator.comb;

categories/euler/prob080-andreoss.pl

@@ -4,7 +4,7 @@
 
 =TITLE Square root digital expansion
 
-=AUTHOR Your name (or nick, if you want)
+=AUTHOR Andrei Osipov
 
 L<https://projecteuler.net/problem=80> 
 

t/categories/euler.t

@@ -396,7 +396,15 @@ subtest {
 
     check-example-solutions($problem, $expected-output, @authors)
 }, "prob063";
+subtest {
+    plan 1;
 
+    my $problem = "prob065";
+    my @authors = <andreoss>;
+    my $expected-output = 272;
+
+    check-example-solutions($problem, $expected-output, @authors)
+}, "prob065";
 subtest {
     plan 1;