How many length-$n$ bitstrings containing $3$ consecutive $0$s …
Mar 10, 2021 · the generating function for the number of strings that do not contain $3$ consecutive $0's$: $$\frac{1-x^3}{1-2x+x^4}$$ The generating functions for the number of strings that do not contain $3$ consecutive zeros and do not contain $4$ consecutive ones : $$\frac{1+2x +3x^2 +3x^3 +2x^4 +x^5}{1-x^2 -2x^3 -2x^4 -x^5}$$
Binary Sequences of length $n$ without $k$ consecutive $1$s or …
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number theory - First 10-digit prime in consecutive digits of e ...
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sequences and series - Gaps between consecutive prime powers ...
As a result, there must exist arbitrarily large gaps between consecutive prime powers. (Added: there are easier ways to prove the density of the prime powers is zero, but this is one official/credible source.)
Why are the differences between consecutive squares equal to the ...
The easiest way to I think about the difference between consecutive powers ( squared cubed or whatever) is using binomial theorem. For example : $(n+1)^4 - n^4 = n^4 + 4n^3 + 6n^2 + 4n + 1 - n^4 = 4n^3 + 6n^2 + 4n + 1$, with the binomial coefficients found by Whatever method you prefer.
Consecutive quadratic residues - Mathematics Stack Exchange
One reference is D. A. Buell and R. H. Hudson, “On runs of consecutive quadratic residues and quadratic nonresidues,” BIT, v. 24, 1984, pp. 243-247. IIRC the proof uses Deligne's well-known bounds on the number of points on curves, but I also know an elementary Ramsey-theoretic argument (which only works for the residue case as opposed to ...
Probability of Consecutive Cards - Mathematics Stack Exchange
Dec 28, 2020 · Let's count the number of hands which don't contain any consecutive cards. If all the cards are distinct and there are no consecutive cards, then we have only two possible hands to consider, and they are $\{1,3,5\},\{2,4,6\}$; there are ${2 \choose 1}{6 \choose 1}^3$ ways to get these hands from our deck of $36$.
parametric - What does it mean by "the consecutive points …
Dec 26, 2018 · The picture shows a curve described by a parametric equation with points marked corresponding to equally spaced values of the parameter, thought of as representing time taken to traverse the curve.
number theory - Existence of Consecutive Quadratic residues ...
Thus, we must have either 2 consecutive residues or 2 consecutive nonresidues. i.e.: 1 and 4 are both residues, so we have R * * R for the quadratic character of 1, 2, 3 and 4. However we fill in the two blanks with Rs or Ns, we will get either 2 consecutive Rs or 2 consecutive Ns. Edited:
combinatorics - Consecutive birthdays probability - Mathematics …
Now since Jan 1 and Dec 31 are consecutive, we need to subtract from this, the number of sets which contain both $\displaystyle 1$ and $\displaystyle k$. This number is same as the number of $\displaystyle d-2$ non-consecutive subsets of $\displaystyle \{3, \dots, k-2\}$ which is $\displaystyle {k-d-1 \choose d-2}$.