Questions tagged [sequences-and-series]
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
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convergence for a series
Show that the series
$$\sum_{n=2}^{\infty}\frac{1}{[\frac{(1+\epsilon)\log n}{\log\log n}]!}$$converges for $\epsilon>0$.
Stirling's approximation gives that the convergence for the series is ...
- 105
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You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?
You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$?
How to ...
- 1,633
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Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?
Edit: "To my serial downvoter: Four downvotes in a row. I wrote the moderators voicing my suspicion it is from one person, and they have confirmed it is indeed just one person. Since you have not ...
- 11.3k
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1
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On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?
I. Recurrences
In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation,
$$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$
within a ...
- 11.3k
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Application of the series reversion
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $a(n)$ be an arbitrary integer sequence such that $a(0)=1$.
Let $b(n)$ be an integer sequence such that
$$b(2^m(2n+1))=\sum\...
- 2,234
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336
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Surprisingly difficult limit of a sequence
Is there an easy way to prove that $|\operatorname{Re}(a_n)| \to \infty$ where $a_n=\left(\frac{1}{2}+i\frac{\sqrt{7}}{2}\right)^n$?
Of course $|a_n| \to \infty$, but we have
$$
\operatorname{Re}(a_n)=...
- 479
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Are the numbers $\sum_{n=1}^\infty\frac1{p(n)}$ and $\sum_{n=1}^\infty\frac1{q(n)}$ transcendental?
For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of ...
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On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?
After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$.
I. Recurrences involving $\zeta(5)$
In Cohen's 2022 paper, ...
- 11.3k
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Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?
I. Two recurrence relations
The first one was also discussed in this MO post. We have the similar,
\begin{align}
(n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
- 11.3k
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Simplification of summation and reverse search
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let $b(n)$ be an integer sequence such that
$$b(2^m(2n+1))=\sum\limits_{k=0}^{m}f(m-k)b(2^kn), b(0)=1$$
Let $s(n,m)$ be an integer ...
- 2,234
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On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence
In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
- 11.3k
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On 12 cfracs: for Catalan's $K$, Gieseking's $\kappa$, and $\pi^2$, $\pi^3$, plus three for $\zeta(3)$ using Zagier's "six sporadic sequences"
I. Some functions
As these will be used in the continued fraction evaluations below, recall the Riemann zeta function $\zeta(s),$ and Dirichlet beta function $\beta(s),$
$$\beta(s) = \sum_{n=1}^\infty\...
- 11.3k
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Sequences that sum up to $\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$
Let $a(n,m)$ be an integer sequence such that
$$a(n,m)=\frac{m^{n+1}}{n+1}\binom{2n}{n}{}_2F_1(1,n+\frac{1}{2}; n+2; -4m(m-1))$$
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$...
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2
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Finding a new level-$7$ pi formula using the relation $j_{7A}(\tau) = \left(\sqrt{j_{7B}(\tau)}+\frac{7}{\sqrt{j_{7B}(\tau)}}\right)^2$
(Note: This third method continues from this post.)
There are level-$7$ pi formulas based on the McKay-Thompson series $T_{7A}$ and Cooper's $s_7$ sequence in this paper. This third method, among ...
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Ramanujan's pi formulas with a twist (nine years later)
(Note: The second method described here continues this post.)
About nine years ago, I made an MO post "Ramanujan's pi formulas with a twist". An answer was informative, but not completely ...
- 11.3k
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1
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Transformations of Ramanujan's 1/pi formulas $\sum_{n=0}^{\infty} s(n)\frac{An+ B}{C^n}$ and Monster moonshine functions
Someone with many papers on Ramanujan's work asked me how I managed to find the closed-forms for the binomial sums of level $10$ in a recent MO post. (A colleague of his wasn't able to find them.) I ...
- 11.3k
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Probability of winning game whereby $T+1$ heads in a row of a coin flip is required to win where $T$ is the number of cumulative tails flipped
I have a weird question which probably seems out of place here but it has proven more difficult than anticipated. I am going to describe the game without showing work toward a solution. Numerically, ...
user504691
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$\frac{m(m+k+1)^n+k}{m+k}$ as closed form for subsequence of the partial sums
Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\frac{m(m+k+1)^n+k}{m+k}$$
There are many sequences in the OEIS that are special cases of a given sequence family:
$a(n,1,1)$ - A007051
$a(n,...
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Does this recurent matrix sequence admit an explicit writing?
I have sequence defined by : 𝐏(n+1)=(𝐈−(Ф.𝐏(n).Ф′+𝐐).𝐇′.(𝐇.(Ф.𝐏(n).Ф′+𝐐).𝐇′+𝐑)^(−𝟏).𝐇).(Ф.𝐏(n).Ф′ +𝐐)
Where :
P(n), Q, R are square, NxN, symmetric, positive semidefinite.
R is square, ...
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1
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333
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On levels $6$ and $10$ of the McKay-Thompson series of the Monster
After the McKay-Thompson series of levels $1,2,3,4$ of the Monster were mentioned in this MO post, levels $6$ and $10$ have very interesting relations as well, and there are unanswered questions for ...
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Sum power series not continuous unit circle
This is (probably) not a research question and I already asked it on StackExchange but I got no answer over there.
Let us consider the sequence $(a_n)_{\geq 1} = \left(\frac{\cos(2\sqrt{2n}+\frac{\pi}{...
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A uniform distribution problem coming from higher dimensions
Thinking about an approximation problem related to random walks, the following question came up.
Suppose we have $m$ numbers $a_1, \ldots, a_m \in \mathbb{R}$ and that $b \in \mathbb{R}$ is not in the ...
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A problem on monotonicity rule for the ratio of two Maclaurin power series
In the paper [1] below, a monotonicity rule for the ratio of two Maclaurin power series was presented as follow.
Let $a_k$ and $b_k$ for $k\in\{0\}\cup\mathbb{N}$ be real numbers and
the power series ...
- 706
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0
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Finding a distance so that this function is a contraction mapping
Let $f(x,y)=(y,\frac{2}{x+y})$ defined on $(0,\infty)\times (0,\infty)$. Is there a distance $d$ on $(0,\infty)\times (0,\infty)$ such that $f$ is a contraction of the metric space $((0,\infty)\times (...
- 479
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2D lattice sum with numerator
I've been struggling a bit with a double sum that arose as the trace of an operator:
$$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$
where $n$ is an even natural number. Is there ...
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Product as closed form for subsequence of the partial sums
Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\prod\limits_{q=0}^{n-1}\sum\limits_{i=0}^{m-1}\sum\limits_{j=0}^{m-i-1}\binom{i+j-1}{j}k^{i+j}q^i$$
Let
$$\ell(n,m)=\left\lfloor\log_m n\...
- 2,234
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0
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On a generalization of A113227 as a subsequence of the partial sums
This question is just a generalization of the one of my previous questions.
Let
$$a(n,m,k)=\sum\limits_{i=1}^{n}u(n,m,k,i)$$
where
$$u(n,m,k,i)=u(n-1,m,k,i-1)+(m-1)(i+k-1)\sum\limits_{j=i}^{n-1}u(n-1,...
- 2,234
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How many digits of $\sqrt{2}$ are known to date?
How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring ...
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Closed form for subsequence of the partial sums of generalized A329369
Let $a(n,m,k)$ be an integer sequence such that
$$a(n,m,k)=\sum\limits_{i=0}^{n}{n\brace i}(m-k)^{n-i}\prod\limits_{j=0}^{i-1}(kj+1)$$
Here ${n\brace k}$ is the Stirling number of the second kind.
...
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Sequences that sum up to Dowling numbers
Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...
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1
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The asymptotic behavior of $F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$, $b>a>0$
Let $b>a>0$. Given $ \lambda>0$, what is the asymptotic behavior of
$$F(\lambda):=\sum_{k=0}^{\infty}\frac{\Gamma{(a k)}}{\Gamma{(b k)}}{\lambda}^{-k}$$
as $\lambda\to 0^{+}$ and as $\lambda \...
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Product-based binary numeration system
I am looking at the following binary numeration system:
$$x =\prod_{k=1}^\infty \Bigg(1+\frac{d_k(x))}{2^k}\Bigg), \quad d_k(x)\in \{0, 1\}.$$
The $d_k$'s are the digits, and $x$ is between $1$ (all ...
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0
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Value of $\pi$ and algorithm for Bernoulli numbers
Chowla and Hartung provide an "algorithm" for computing Bernoulli numbers in this paper.
In particular, if the Bernoulli numbers are defined by
$$\frac{x}{e^x-1}=1-\frac{x}{2}+\sum_{n=1}^\...
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0
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Sequences that sum up to the many sequences in the OEIS
Let
$$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$
Here square brackets denote Iverson brackets.
There are many sequences in the OEIS that are ...
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Laurent Series $\sum_{n=-1}^\infty a_n x^n$ when $a_{-1} = \infty$
When dealing with complex functions, if $f(x)$ has a simple pole, then we can find the coefficient $a_{-1}$ in the Laurent expansion $f(x) = \sum_{n=-1}^\infty a_n x^n$ by evaluating the limit $\lim_{...
- 1,485
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0
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Factor group of all the sequences by the subgroup of bounded sequences
Consider the group G of the sequences of real numbers (the group operation is addition). It contains a subgroup H of bounded sequences.
Is there any nice description of the factor group G/H ?
It is ...
- 4,661
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Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$
Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$.
The sequence begins with
$$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
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1
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Does rapid convergence of the Cesaro sums imply convergence of the original sequence?
Question: Let $a_n$ be a sequence of real numbers. Is it true that for every $\varepsilon > 0$, if
$$\left \lvert \frac{1}{N} \left ( \sum_{n=0}^{N-1} a_n \right )\right \rvert < \frac{1}{N^{1+\...
- 2,574
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1
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Where does the Weierstrass expansion of $\operatorname{sn}$ come from?
In Table of Integrals, Series and Products (p. 869) by Gradshteyn and Ryzhik, I found the identity (called 'the Weierstrass expansion of $\operatorname{sn}$')
$$\operatorname{sn}u=\frac{B}{A}$$
where $...
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0
answers
63
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Integer coefficients such that $T(n,k)=R(n,k)-R(n,k-1)$
Let $a(n)$ be A000085, i.e., the number of self-inverse permutations on $n$ letters, also known as involutions; number of standard Young tableaux with $n$ cells. Here
$$a(n) = a(n-1) + (n-1)a(n-2), a(...
- 2,234
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1
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If $f(x) = \sum_{n=0}^\infty a_n x^n$, then $\int_{-\infty}^\infty f(x^2) dx = \pi i a_{-\frac{1}{2}}$
I've noticed a curious relationship between the coefficient $a_n$ for a power series and the integral of the real line. For instance, take $f(x) = e^{-x} = \sum_{n=0}^\infty \frac{(-1)^n}{n!} x^n$. ...
- 1,485
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vote
0
answers
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Recurrence for the number of permutations with a given excedance set
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...
- 2,234
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0
answers
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Recurrence for the A284005
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...
- 2,234
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votes
0
answers
66
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Permutation that produces permutations
Let $f(n)$ be A000045, i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, 3, ...
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1
answer
98
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proving inequality in Riemann zeta function
Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}. $$ When this ...
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0
answers
72
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Uniqueness of the permutation
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 2,234
0
votes
0
answers
62
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Recurrence relation involving the $\max$ function (reference request): $p_{n + 1} = p_{n} + \max\{r_{p_{n} - 1} - 1, r_{p_{n}}\}$
Consider two sequences $(p_{n})_{n\in\mathbb{N}}$ and $(r_{n})_{n\in\mathbb{N}}$ such that they satisfy the following recurrence relation:
\begin{align*}
p_{n + 1} = p_{n} + \max\{r_{p_{n} - 1} - 1, ...
- 161
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2
answers
196
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Continuous functions on $[0,1]^\omega$ and a product lower bound
I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology).
The map $f:X\to [0, 1]$ given by $(x_i)\mapsto \prod x_i$ is well-defined and Borel but not ...
- 53
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0
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119
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Ask for a proof of an inequality involving the Bernoulli numbers
Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
- 706
7
votes
3
answers
507
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Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distributed in $[0,1)$?
A classical theorem of Kronecker says that the sequence $(\{\alpha_1 n\}, \{\alpha_2 n\},\dots,\{\alpha_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that $1,\alpha_1,\...
- 1,490