Intereting Posts

Finite rings without zero divisors are division rings.
converse to the jordan curve theorem
Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$
Countable closed sets
Fibonacci( Binet's Formula Derivation)-Revised with work shown
Product of matrices of different order is not invertible
Generalizing Ramanujan's sum of cubes identity?
an intriguing integral $I=\int\limits_{0}^{4} \frac{dx}{4+2^x} $
Find the upper bound of the derivative of an analytic function
Disjoint compact sets in a Hausdorff space can be separated
Evaluate this and also the indefinite case
Endpoint of a line knowing slope, start and distance
Poincare duality in group (co)homology
Fractional part of rational power arbitrary small
Prove $\ker {T^k} \cap {\mathop{\rm Im}\nolimits} {T^k} = \{ 0\}$

I want a sequence that alternates between being an even integer and being an odd integer and I’ve come up with this sequence $ s_n=\lfloor \frac{n}{2} \rfloor $. So, it goes $0,1,1,2,2,3,3,\ldots$ and I was wondering if I can do something similar without using a floor or ceiling function. The important thing is that it goes back and forth between even and odd without simply alternating.

- Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$
- Euler's transformation to derive that $\sum\limits_{n=1}^{\infty}\frac{1}{n^2}=\sum\limits_{n=1}^{\infty}\frac{3}{n^2\binom{2n}{n}}$
- The limit of a sequence $\lim_{n\rightarrow \infty}\prod_{k=0}^{n-1}( 2+\cos \frac{k\pi }{n})^{\pi/n}$.
- Number of strings lenght $n$ with no consecutive zeros
- $1 + 1 + 1 +\cdots = -\frac{1}{2}$
- Closed formula for the sum of the following series
- Use Fourier series for computing $\sum_{n=1}^{\infty}\frac{1}{(2n-1)^2}$
- $\zeta(2)=\frac{\pi^2}{6}$ proof improvement.
- Evaluating the series $\sum_{n=1}^{\infty} \frac{1}{n^{3} \binom{2n}{n}} $
- General question on relation between infinite series and complex numbers

Well, one way to do it is $$ s_n = \frac{n + \frac{(-1)^n – 1} 2} 2 = \frac n 2 + \frac {(-1)^n} 4 – \frac 1 4. $$

This yields the exact same sequence as your formula: $0, 1, 1, 2, 2, 3, 3, \dotsc$

The way it works is that $\frac{(-1)^n – 1} 2$ equals $0$ if $n$ is even and $-1$ if $n$ is odd. Adding this to $n$ subtracts $1$ from each odd term, giving the sequence $0, 2, 2, 4, 4, 6, 6, \dotsc$, and dividing this by $2$ then produces your sequence.

The sum of the first $n$ integers, giving $n(n+1)/2$, is a simple example: $$1,3,6,10,15,21,28,36,45,55,\ldots$$

How about something like this?

$$F_n = F_{n-1} + F_{n-2} + F_{n-3}$$

where $F_1 = 0, F_2 = F_3 = 1$

You can prove that the series goes odd, odd, even, even, etc by adding in mod 2.

- How to prove that $\lim_{n \to\infty} \frac{(2n-1)!!}{(2n)!!}=0$
- How to write this statement in mathematical notation?
- Expression from generators of Special Linear Groups II
- What is the minimum $ \sigma$-algebra that contains open intervals with rational endpoints
- How to prove that a subspace is in a vector space?
- Example of torsion-free module
- The diophantine equation $x^2+y^2=3z^2$
- Two non-homeomorphic spaces with continuous bijective functions in both directions
- Finding XOR of all subsets
- Deriving Taylor series without applying Taylor's theorem.
- Group of Order $5$
- How can I improve my problem solving/critical thinking skills and learn higher math?
- If $p\equiv 1,9 \pmod{20}$ is a prime number, then there exist $a,b\in\mathbb{Z}$ such that $p=a^{2}+5b^{2}$.
- How can this integral expression for the difference between two $\zeta(s)$s be explained?
- The area of the superellipse