Questions tagged [recurrence-relations]
Questions regarding functions defined recursively, such as the Fibonacci sequence.
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If 2 recursion relation has the same coeficients are the solution proportional?
In the book of sakurai
Modern Quantum Mechanics they have this $$
\begin{aligned}
& \sqrt{(j \mp m)(j \pm m+1)}\left\langle j_1 j_2 ; m_1 m_2 \mid j_1 j_2 ; j, m \pm 1\right\rangle \\
& =\sqrt{...
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Got stalled with a simple sequence problem: $x_1 = a,ドル 0ドル<a<1$; $x_{n+1} = \ln(1+x_n)$; $\lim_{n\to \infty} nx_n$; [closed]
Define $\{x_n\}$ sequence this way:
$$x_1 = a, \quad 0<a<1 $$
$$x_{n+1}=\ln(1+x_n)$$
$$\lim_{n\to \infty}nx_n \rightarrow \;?$$
While it’s not hard to prove $\lim_{n\to \infty}x_n=0,ドル I still ...
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Closed form of Homogeneous Linear Recurrence Relations
I know that given a linear homogeneous recurrence relation of order k:
$$a_n = c_1 a_{n-1} + c_2 a_{n-2} + \cdots + c_k a_{n-k}$$
We can get the characteristic equation:
$$r^n = c_1 r^{n-1} + c_2 r^{n-...
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How to calculate the iterations of Fibonacci Sequence under square roots
Compute the value of
$$\sqrt{1 + F_2\sqrt{1 + F_4\sqrt{1 + F_6\sqrt{1 + F_{2n}\ldots}}}}$$
where $F_n$ denotes the $n$-th Fibonacci number with $F_0 = 0,ドル $F_1 = 1$.
This is a problem from a sheet ...
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A hidden modular order in a bizarre recurrence involving gcd
Let’s play with the recurrence (see also this post of mine)
$$x_{n+1} = \frac{x_n + x_{n-1}}{\gcd(x_n + x_0x_1,\; x_n + x_{n-1})},
\qquad (x_0, x_1)\in\mathbb Z^2.$$
At first sight it looks chaotic: ...
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A family of symmetric recurrent sequences
Let's consider the family of symmetric recurrent sequences
$$kx_n+c=x_{n+k-1}+x_{n-k+1}$$
where $k\gt0$ and $c\ge0$ are integers. Also we define $d=k-1,ドル so we can write the recurrence $R(d,k,c)$
$$x_{...
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Is it possible to solve recurrence relation with non-constant coefficient when coefficient is not explicitly given in terms of $n$?
I have a recurrence relationship as:
$$F_{n}=a_nF_{n+1}+a_nF_{n-1}$$
Is it possible to solve such a relation (using a generating function) when the explicit value of $a_n$ is given but not explicitly ...
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A new prime-indexed additive sequence and its properties
I would like to share and ask about the following integer sequence that I have been experimenting with, while looking for "simple but rich" sequences in the spirit of OEIS contributions.
I start from $...