Recent content by alec_tronn

Alright, using the Apart function in Mathematica, I separated the generating function into: [-1/(8(-1+x)^3)] + [1/(4(-1+x)^2)]- [9/(32(-1+x))]+ [1/(16(1+x)^2)]+ [5/(32(1+x))]+ [(1+x)/(8(1+x^2))] Then, I turned those each into the followings infinite series...

Homework Statement Determine the number of solutions in nonegative intergers to the equation: a + 2b + 4c = 10^{30} Homework Equations The generating function I've found is f(x) = 1/[(1-x^{4})(1-x^{2})(1-x)] The Attempt at a Solution I'm pretty sure I need to get from here to an...

Since the Fibonacci sequence is 0,1,1,2,3,5,8,... the Fibonacci Cubed sequence is 0,1,1,8,27,125,512... it's just cubing each element of the Fibonacci sequence. Upon further research, I found that that rule was found and proven by Zeitlin and Parker and published in the Fibonacci Quarterly in...

Homework Statement Find and prove the recurrence relation for the Fibonacci cubed sequence. Homework Equations By observation (blankly staring at the sequence for an hour) I've decided that the recurrence relation is G_{n} = 3G_{n-1} + 6G_{n-2} - 3G_{n-3} - G_{n-4} (where G is Fibonacci...

Thanks! Alright, using the ball\box analogy, I got: All 4 in 1 box -> 13 choose 1 3 in 1, 1 in 1 -> (13 choose 1)*(12 choose 1) 2 in 1, 2 in another 1 -> 13 choose 2 2 in 1, 1 in 1, 1 in another 1 -> (13 choose 1)*(12 choose 2) 1 ball in each box for some 4 box -> (13 choose 4), I...

I considered that. Using that method I tried (13 choose 1)^{4} (choosing 1 topping for each of your 4 spaces), and then divided that by 4! (because order isn't important) and I got a decimal (1190.0416666666666666666666666667). Am I doing that wrong?

Homework Statement You can order a pizza with up to four toppings (repetitions allowed) from a set of 12 toppings. The order of the toppings is unimportant. How many different pizzas can you order? (To clarify: this UP TO four toppings, so a completely empty pizza is fair game, as is a pizza...

The thing is... the antecedent isn't true. So if you were asked to prove that "If there exists integers m and n such that 12m+15n=1 then m and n are negative" all you'd do is prove that the antecedent is false and say "by default this statement is true."...

Homework Statement Prove that the number of permutations p on the set {1,2,3,...,n} with the property that |p(k)-k| \leq 1, for all 1\leqk\leqn, is the fibonacci number f_{n} The Attempt at a Solution I guess I don't understand what it's asking. I thought I knew what a permutation was...

That's all the question is trying to get at... in a statement P->Q, if the P part is false, the entire statement is true by default. This is chapter 1, don't over think it too much. Just like Dick said "If there exist integers m and n such that 12m+15n=1, then m and n are bananas" is also a very...

I'm not sure induction would be of much use here unless I knew the algebraic sequence to go through to make the two sides equal. But it did give me a little more insight... I tried to do the first couple k's, so that I'd have a base case to start from: k=1, (kn)! = (1n)! which is...

Show that (kn)! is divisible by (n!)^k ? Homework Statement Show that (kn)! is divisible by (n!)^k. The Attempt at a Solution I attempted a factorization of the problem as follows: (kn)! = (kn)*(kn-1)*(k-2)...(kn-n+1) * ((k-1)n-1)*((k-1)n-2)*((k-1)n-1)...((kn-1)n-n+1)*...

Oooooooh. Okay Thanks a lot. I got it now.

I think I'm following you, but I have a couple questions. Physically, yes bouncing off of each other is the same as passing through, but then the names for the particles have switched, and we're required to have the original particles in their original positions, so how to I know\prove that...

Homework Statement Suppose that five particles are traveling back and forth on the unit interval [0,1]. Initially, all five particles move to the right with the same speed. (The initial placement of the particles does not matter as long as they are not at the endpoints.) When a particle...