0
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0answers
14 views

Comparing Generalized Continued Fractions

Gosper lays out a method (under Approximations) for comparing regular (a.k.a simple) continued fractions which have all partial numerators set to 1. Continue comparing terms until they differ, then ...
1
vote
2answers
29 views

Finding the inhomogeneous solution

$x_{n+2} = x_{n+1} + 20x_n + n^2 + 5^n \text{ with } x_0 = 0 \text{ and } x_1 = 0$ How would I find the inhomogeneous solution for this since the homogenous solution is 0 given initial conditions?
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1answer
37 views
0
votes
0answers
49 views

Tight bounds for the number “2 in a hexagon” wanted (Steinhaus-Moser-Notation)

The Steinhaus-Moser-function is defined in the following way : $$M(n,1,3) = n^n$$ $$M(n,1,p+1) = M(n,n,p)$$ for all $p\ge3$ $$M(n,m+1,p) = M(M(n,1,p),m,p)$$ for all $p\ge3$ and $m\ge1$ The ...
1
vote
1answer
101 views

$n$-Bit Strings Not Containing $010$

So, I am asked to consider the number of $n$-bit strings that don't contain $010$ by considering the following $m$-leading-zero cases for $m\geq 0$, where $m\in \mathbb{N}$: $1\cdots$ $01\cdots$ ...
0
votes
1answer
111 views

Paths Within a Lattice

So, I'm reading this proof: Lemma 4.2. The Schröder numbers $(r(n):n\geq0))$ satisfy $$r(n)=r(n-1)+\sum_{k=0}^{n-1}r(k)r(n-1-k)\text{ for }n\geq1,\text{ with } r(0)=1$$ Proof. The Schröder number ...
3
votes
1answer
190 views

constructive proof of the infinititude of primes

There are infinitely many prime numbers. Euclides gave a constructive proof as follows. For any set of prime numbers $\{p_1,\ldots,p_n\}$, the prime factors of $p_1\cdot \ldots \cdot p_n +1$ do not ...
4
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0answers
156 views

Fractional iteration of the Newton-approximation-formula: how to resolve one unknown parameter?

For my own exercising I tried to find a closed form expression for the Newton-approximation algorithm, beginning with the simple example for getting the squareroot of some given $ \small z^2 $ by ...