All Questions
Tagged with integer-sequences polynomials
19 questions
19
votes
1
answer
1k
views
Is OEIS A007018 really a subsequence of squarefree numbers?
A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims
Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004
Is that really so?
As far as I know, it is an open ...
- 25.4k
16
votes
2
answers
1k
views
are these polynomials or rationals functions?
Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by
$$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$
...
- 43.2k
9
votes
2
answers
546
views
Can you tie up these Laurent sequences?
Fix an integer $k\geq3$. Define the two families of sequences $\{x_n\}$ and $\{y_n\}$ according to the rules:
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2+\cdots+x_{n-k+1}^2}{x_{n-k}} \qquad n\geq k$$
and
$$y_n=\...
- 43.2k
8
votes
0
answers
1k
views
Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?
The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
- 13.9k
6
votes
1
answer
402
views
Values of the determinants $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}\ (m=1,2,3,\ldots)$
For positive integers $m$ and $n$, let $D_m(n)$ denote the determinant $\det[(j-k)^m+\delta_{jk}]_{1\le j,k\le n}$, where the Kronecker delta $\delta_{jk}$ is $1$ or $0$ according as $j=k$ or not.
...
- 15.6k
6
votes
1
answer
315
views
Integral points of polynomials - a Furstenburg-type "topology" on $\mathbb{Z}$
Given $S \subseteq \mathbb{C}$, define $\displaystyle \mathfrak{c}(S) = \bigcap_{p(x) \in \mathbb{C}[x] \wedge p(S) \subseteq \mathbb{Z}}p^{-1}(\mathbb{Z}) \supseteq S$ ("the integral points ...
- 1,543
6
votes
2
answers
389
views
Conjectured Somos-like closed form of recurrences with polynomial coefficients
From Our short paper
For polynomial $F$ with integer coefficients, define the recurrence
$f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that
$f(n)$ satisfy Somos like sequence
$f(n)=\frac{G(f(n-1)...
- 25.4k
6
votes
0
answers
385
views
A strange polynomial equality
In my answer to this question, I have obtained that the polynomial $p(x)$ of degree $2n$ with nonnegative values on $[-1,1]$ with $p(\pm1)=1$ has $\int_{-1}^1 p(x)\,dx\geq \frac{4}{(n+1)(n+2)}$, and ...
- 23.7k
5
votes
1
answer
345
views
Why does this "factorial sequence" appear in the OEIS?
For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$
I ...
- 3,499
5
votes
0
answers
183
views
On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
4
votes
0
answers
302
views
Identities for powers of functions based on generalization of Lagrange interpolation
Lagrange polynomial can be used to obtain an identity:
$$(k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n \frac{t-d_j}{d_i-d_j},$$
which holds for any integer $n>0$, any real ...
- 34.4k
2
votes
1
answer
281
views
Curious sequences of polynomials
Given an integer $k\geq 2$, and $k+1$ invertible initial
values $s_0,s_1,\ldots,s_k$ in some commutative ring $\mathcal A$
we set
$$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{...
- 17.9k
2
votes
1
answer
196
views
Guess (or upper bound) the general formula for a double sequence
Let $t,s \geq 0$ be integers. We have the following recursive formula:
$$f(t+1,s) = f(t,s) + f(t,s-1) + \sum_{0\leq a,b,c \leq h(t):\\a+b+c = s-1}f(t,a)f(t,b)f(t,c),$$ where
$$h(t) = \frac{1}{2}3^t -\...
- 515
2
votes
2
answers
210
views
An identity for the ratio of two partial Bell polynomials
Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that
the ...
- 1,101
1
vote
1
answer
594
views
Polynomials, $3^x$ and the Collatz conjecture
$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open.
Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...
- 13.9k
1
vote
0
answers
73
views
On a type of equations that involve certain multiplicative functions and polynomials, in relation to their number of solutions
Past weekend I was interested in the sequence A058891 from the On-Line Encyclopedia of Integer Sequences, from this, inspired by the equation due to Benoit Cloitre (2002) that shows the comments, I ...
0
votes
0
answers
62
views
Linear recurrences in coefficients of powers of quotients of polynomial rings
It is known that linear recurrences with constant coefficients
can be computed via powers in $\mathbb{Z}[x]/f(x)$.
We believe that this generalizes to quotients of multivariate polynomial
rings.
Let $...
- 25.4k
0
votes
0
answers
115
views
Roots of a family of 4-parameter polynomials
Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by
$$
f(x)=x^q-kx^{q-p}-\ell.
$$
This polynomial is related to a family of two-...
- 29
0
votes
0
answers
86
views
Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce square-free integers
Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians to get prime constellations or statements related to primality tests for these ...