P106: Added Java, Python, Mathematica, Haskell solutions.

Author: nayuki

Answers.txt

@@ -110,6 +110,7 @@ Problem 102: 228
 Problem 103: 20313839404245
 Problem 104: 329468
 Problem 105: 73702
+Problem 106: 21384
 Problem 107: 259679
 Problem 108: 180180
 Problem 109: 38182

haskell/p106.hs

@@ -0,0 +1,16 @@
+{- 
+ - Solution to Project Euler problem 106
+ - Copyright (c) Project Nayuki. All rights reserved.
+ - 
+ - https://www.nayuki.io/page/project-euler-solutions
+ - https://github.com/nayuki/Project-Euler-solutions
+ -}
+
+import qualified EulerLib
+
+
+setSize = 12
+main = putStrLn (show ans)
+catalan n = div (EulerLib.binomial (n * 2) n) (n + 1)
+ans = sum [(EulerLib.binomial setSize (i * 2)) * ((div (EulerLib.binomial (i * 2) i) 2) - (catalan i))
+	| i <- [2 .. (div setSize 2)]]

java/EulerTest.java

@@ -117,6 +117,7 @@ public final class EulerTest {
 	@Test public void testP103() { assertEquals("20313839404245"          , new p103().run()); }
 	@Test public void testP104() { assertEquals("329468"                  , new p104().run()); }
 	@Test public void testP105() { assertEquals("73702"                   , new p105().run()); }
+	@Test public void testP106() { assertEquals("21384"                   , new p106().run()); }
 	@Test public void testP107() { assertEquals("259679"                  , new p107().run()); }
 	@Test public void testP108() { assertEquals("180180"                  , new p108().run()); }
 	@Test public void testP109() { assertEquals("38182"                   , new p109().run()); }

java/p106.java

@@ -0,0 +1,32 @@
+/* 
+ * Solution to Project Euler problem 106
+ * Copyright (c) Project Nayuki. All rights reserved.
+ * 
+ * https://www.nayuki.io/page/project-euler-solutions
+ * https://github.com/nayuki/Project-Euler-solutions
+ */
+
+import java.math.BigInteger;
+
+
+public final class p106 implements EulerSolution {
+	
+	public static void main(String[] args) {
+		System.out.println(new p106().run());
+	}
+	
+	
+	private static final int SET_SIZE = 12;
+	
+	public String run() {
+		BigInteger ans = BigInteger.ZERO;
+		for (int i = 2; i * 2 <= SET_SIZE; i++) {
+			BigInteger x = Library.binomial(SET_SIZE, i * 2);
+			BigInteger y = Library.binomial(i * 2, i).shiftRight(1);
+			BigInteger z = Library.binomial(i * 2, i).divide(BigInteger.valueOf(i + 1));  // Catalan number
+			ans = ans.add(x.multiply(y.subtract(z)));
+		}
+		return ans.toString();
+	}
+	
+}

mathematica/p106.mathematica

@@ -0,0 +1,12 @@
+(* 
+ * Solution to Project Euler problem 106
+ * Copyright (c) Project Nayuki. All rights reserved.
+ * 
+ * https://www.nayuki.io/page/project-euler-solutions
+ * https://github.com/nayuki/Project-Euler-solutions
+ *)
+
+setSize = 12;
+CatalanNum[n_] := Binomial[n * 2, n] / (n + 1)
+Sum[Binomial[setSize, i * 2] * (Binomial[i * 2, i] / 2 - CatalanNum[i]),
+  {i, 2, Floor[setSize / 2]}]

python/eulertest.py

@@ -129,6 +129,7 @@ def main():
 	103: "20313839404245",
 	104: "329468",
 	105: "73702",
+	106: "21384",
 	107: "259679",
 	108: "180180",
 	109: "38182",

python/p106.py

@@ -0,0 +1,24 @@
+# 
+# Solution to Project Euler problem 106
+# Copyright (c) Project Nayuki. All rights reserved.
+# 
+# https://www.nayuki.io/page/project-euler-solutions
+# https://github.com/nayuki/Project-Euler-solutions
+# 
+
+import eulerlib
+
+
+def compute():
+	SET_SIZE = 12
+	
+	def catalan(n):
+		return eulerlib.binomial(n * 2, n) // (n + 1)
+	
+	ans = sum(eulerlib.binomial(SET_SIZE, i * 2) * (eulerlib.binomial(i * 2, i) // 2 - catalan(i))
+		for i in range(2, SET_SIZE // 2 + 1))
+	return str(ans)
+
+
+if __name__ == "__main__":
+	print(compute())