1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Number of Sequences

  1. Mar 4, 2008 #1
    1. The problem statement, all variables and given/known data

    In how many ways can we form a sequence of non-negative integers a_1,a_2,...,a_(k+1) such that the difference between the successive terms is 1 (any of them can be bigger) and a_1 =0.

    2. Relevant equations

    3. The attempt at a solution

    For k=1, there is only one way: 01. For k=2, there are two ways: 012 or 010. For k=3, there are three ways: 0121,0123 or 0101. For k=4 there are six ways: 01210,01212,01232,01234,01010 or 01012. Note that when k is even, the number of such sequences is twice the number of such sequences when we have k-1 because when k is odd, there is no possibility for the sequence to finish with 0, and so we can add both of the two possible choices to the end of the sequence. But i couldn't find a nice formula relating the case k is odd to the one k is even? Can you help me with this?

    Note: There are 10 ways for k=5, 20 ways for k=6 and 35 ways for k=7.
  2. jcsd
  3. Mar 4, 2008 #2
    I don't think there's a way to describe both in the same way. The conditions for creating new sequences is slightly different depending on whether k is odd or even (as you realized).
    Last edited: Mar 4, 2008
  4. Mar 11, 2008 #3
    The results of this problem are quite interesting to me. When k is an even integer, say 2n, then the number of such ways is C(2n n); and when k is an odd integer, say 2n+1, the number of such ways is C(2n+1 n). Also i found that for k odd, say 2n+1, the number of ways to realize our aim is [(2*the number of such ways for k=2n) - C_n],where C_n denotes the Catalan number, I couldn't find a nice way to prove the above claim (which is obviously true). Can you demonstrate why these are true?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Number of Sequences
  1. Number sequences hard (Replies: 6)

  2. Number sequence (Replies: 1)