Number of Sequences with Diff. of 1 and a_1=0

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In summary, the problem is to find the number of ways to form a sequence of non-negative integers a_1, a_2, ..., a_(k+1) such that the difference between successive terms is 1 and a_1 = 0. For k = 1, there is only one way, for k = 2, there are two ways, and for k = 3, there are three ways. When k is even, the number of ways doubles compared to when k is odd. For k = 4, there are six ways, and for k = 5, there are 10 ways. The pattern continues with 20 ways for k = 6 and 35 ways for k =
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Homework Statement



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.


Homework Equations





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.
 
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  • #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).
 
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  • #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?
 

Related to Number of Sequences with Diff. of 1 and a_1=0

1. What is the definition of "Number of Sequences with Diff. of 1 and a_1=0"?

The number of sequences with a difference of 1 and a first term of 0 is the total number of possible sequences where each term is 1 more than the previous term and the first term is 0.

2. How is the "Number of Sequences with Diff. of 1 and a_1=0" calculated?

The formula for calculating the number of sequences with a difference of 1 and a first term of 0 is 2^(n-1), where n is the number of terms in the sequence.

3. Can you provide an example of a sequence with a difference of 1 and a first term of 0?

One example of a sequence with a difference of 1 and a first term of 0 is: 0, 1, 2, 3, 4, 5, ...

4. How does the number of terms in the sequence affect the "Number of Sequences with Diff. of 1 and a_1=0"?

The number of terms in the sequence directly affects the "Number of Sequences with Diff. of 1 and a_1=0". As mentioned before, the formula for calculating this number is 2^(n-1), so the more terms there are in the sequence, the greater the total number of possible sequences.

5. Is the "Number of Sequences with Diff. of 1 and a_1=0" relevant in any specific field of science?

Yes, this concept is often used in mathematics and computer science, particularly in the study of number sequences and algorithms. It also has applications in physics, biology, and other fields where pattern recognition is important.

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