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Questions tagged [integer-sequences]

For questions about sequences of integers. References are often made to the online resource oeis.org.

0
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1answer
298 views

A possible surprise involving Euler's constant $e$ [closed]

Let \begin{align*} c_n &= n!\left(e-\sum_{k=0}^n \frac{1}{k!}\right) \\ \\ u_n &= \bigg\lfloor{\frac{1}{c_n} \bigg\rfloor} \\ \\ v_n &= \bigg\lfloor{\frac{1}{1/c_n-\lfloor{u_n} \rfloor}} ...
18
votes
1answer
484 views

Order of Conway's “look and say” recurrence

Let $L_n$ be the length of the $n$th term of Conway's "look and say" sequence (https://oeis.org/A005341). The generating function $F(x)= \sum_{n\geq 0}L_nx^n$ is a rational function, say $P(x)/Q(x)$ ...
6
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0answers
203 views

Is the permanent of the matrix $[(\frac{i+j}{2n+1})]_{0\le i,j\le n}$ always positive?

Recall that the permanent of an $n\times n$ matrix $A=[a_{i,j}]_{1\le i,j\le n}$ is defined by $$\operatorname{per}A=\sum_{\sigma\in S_n}\prod_{i=1}^n a_{i,\sigma(i)}.$$ In 2004, R. Chapman [Acta ...
3
votes
0answers
101 views

Inequalities about tripling and doubling sumsets

Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following: Question 1:...
0
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0answers
93 views

Ulam Sequence and Primes

The Ulam sequence is defined as 1,2,3,4,6,8,11,... where, after 2, a number is added to the sequence if and only if it is expressible as a sum of two distinct preceding numbers in a unique way. It ...
12
votes
1answer
239 views

Determinant of a matrix filled with elements of the Thue–Morse sequence

Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be ...
3
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0answers
95 views

Generating a Penrose tessellation around a given tile

Given a starting Penrose tile, I need to build a "spiraling" tessellation around it. The following picture illustrates the request: In this example, the starting tile is a "thin rhombus" (the pink ...
29
votes
1answer
2k views

A remarkable almost-identity

OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$. Mikhail Kurkov noticed that it ...
2
votes
1answer
170 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 -\...
0
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0answers
79 views

Infinite difference length of integer subsets

Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition: We say $A$ has infinite difference length, if (a) For every integer $n$ there exist a ...
7
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0answers
190 views

Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history

Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces. Counting the ...
4
votes
1answer
325 views

The range of the Euler totient function and multiplication by 28

If $n$ is in the range of the Euler totient function, certain multiples of $n$ are likewise guaranteed to be totient values. The simplest nontrivial example of this is that, if $n$ is in the range of ...
-4
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1answer
135 views

Covering system of congruences with specific properties?

A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$, ($r\geq2$) is called a covering system of congruences if every integer belongs to at least one of the residue classes, ...
1
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1answer
72 views

constructing a covering system of congruences?

A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, ...
1
vote
1answer
140 views

How many points appear in the plane when the chain of n-gons is close?

Let $A_{11}A_{12}\cdots A_{1n}$ be a regular $n$ polygon, we call $A_{11}A_{12}\cdots A_{1n}$ is the $1st-n-gons$. Now we construct the $2nd-n-gon$ based two condition as follows: $2nd-n-gons$ is ...
-1
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1answer
65 views

Create approximations of finite integer sequence

Given a function of real numbers f(x), I can create approximations to arbitrary precision using Taylor polynomials. Is there something equivalent in the discrete case when I have a sequence of ...
5
votes
1answer
224 views

Additive basis of order 2

Can we find $\alpha>1$ such that $u=(\lfloor n^\alpha\rfloor)_{n\geqslant0}$ is an additive basis of order $2$ (i.e. $\forall x\in\mathbb{N}, \exists(n,m)\in\mathbb{N}^2, x=u_n+u_m$) ? Remark : ...
4
votes
1answer
139 views

An inequality involving $k$-generalized Fibonacci numbers

I have worked on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, to solve completely the equation I have one complicated case and I proved ...
4
votes
2answers
247 views

Periods of natural numbers

Define a function $F$ on the natural numbers $\geq 2$ as follows: Start with $a \geq 2$ and let $b$ be the smallest prime divisor of $a$ and $c:=a+b$ and let $d$ denote the largest prime divisor of $c$...
8
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4answers
849 views

Find a formula for the recurrent sequence $q_{n+1}=q_n(q_n+1)+1$

Find an analytic formula for the recurrent sequence $$q_{n+1}=q_n(q_n+1)+1,\;\;q_0\in\mathbb N.$$ (The question was asked on 03.05.2018 by M. Pratsovytyi, see page 109 of Volume 1 of the Lviv ...
4
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0answers
64 views

Closed form for integer series from enumerative geometry problem?

Is there a closed form for the following integer sequence: $$ 1,6,145,8806,830622,100317140,14342519633,2325250316950,... $$ This is the degree of the $2n$-th power of the Schubert class $\sigma_{2,...
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0answers
109 views

Does each prime $p>3$ have a quadratic nonresidue which is a Mersenne number?

Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime. QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?...
2
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0answers
97 views

Sieving the values of an arithmetic sequence which is infinitely many times $1$

I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set: $$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
0
votes
1answer
141 views

The Euler's totient function and the product of distinct primes dividing $n$ versus the Heronian means

For integers $n\geq 1$ with $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p$$ we denote the squarefree kernel or radical of an integer $n$ (see if you want this Wikipedia). And $\...
3
votes
1answer
291 views

Is there a better proof for this than using the 10-adic numbers?

Here are two somewhat strange sums using the shifted decimal forms of the powers of $3.$ $\begin{equation*}\begin{array}{ccccccc} &1&&&&&& \\ &&3&&&&...
0
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0answers
182 views

Cancellations in random weighted sum

Fix a set of integer weights $h_1,\dots,h_{2t}$ with $|h_i|\in(n,\alpha\cdot n)$ for $\alpha\in(1,2)$ with $t$ of $h_i$'s being positive and $t$ being negative. Choose a random vector of variables $...
0
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1answer
326 views

Is there any number other than 109 whose reciprocal contains the Fibonacci sequence? [closed]

Let $p$ be any odd number, and compute $1/p$ to $p$ decimal places. Compare your answer with the string that is formed by appending all remainders of $(10^n\ \text{mod p}) \text{ mod p}$ where ${0 <...
2
votes
1answer
642 views

Power tower made of $2$s and $3$s: too high, too soon?

A power tower of a number $x$ is typified by $$ x^{x^{x^{x^{x^{x^{x^{x^{x^x}}}}}}}}.$$ Here, however, we take the liberty of referring to the set $T$ of "$\{2,3\}$-power towers"; i.e., numbers $$...
5
votes
1answer
248 views

Simply generated sequences with mysterious differences

Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$...
20
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2answers
1k views

A possibly surprising appearance of $\sqrt{2}.$

Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
5
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0answers
218 views

Elliptic curve sequences needed for universal forgery

Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation $$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$ where $k$ is unknown, $f_{k}...
1
vote
1answer
181 views

An elementary sequence question [closed]

Below is a problem, from an old Silk Road olympiad. Define an infinite sequence, $a(n)$, such that, $a(1)=a(2)=1$; $$ a(n)=a(a(n-1))+a(n-a(n-1)),\forall n\geq 3. $$ Show that, for every $n\geq 1$, $a(...
7
votes
2answers
353 views

Limit associated with complementary sequences

Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
29
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7answers
3k views

Examples of integer sequences coincidences

For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
0
votes
1answer
101 views

Elementary description to count of perfect squares - II

What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$? Is $g(a)=1\iff a=b^2+...
0
votes
1answer
117 views

Elementary description to count of perfect squares - I

Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...
2
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0answers
45 views

Collinearity in Enumerations of the Rationals

I am looking for a solution of the No Three-in-a-Line problem for the whole $\mathbb{Z}\times\mathbb{Z}$ plane and had the idea, to use a non-redundant enumeration of the rationals, like the breadth-...
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votes
1answer
172 views

What are the patterns of the sequence of polynomials? [closed]

In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them): \begin{align} & f(2) = 1+t, \\ & f(3) = 1+4t+3t^2, \\ & f(4) = 1+6t+12t^2+7t^3, \\ &...
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0answers
119 views

A provably infinite infinitary aliquot sequence?

A divisor of n is called infinitary if it is a product of divisors of the form $p^{y_a 2^a}$, where $p^y$ is a prime power dividing n and $\sum_a y_a 2^a$ is the binary representation of y. [from OEIS ...
1
vote
2answers
459 views

Can these sequences stay integer-valued as many times as we want and then fail?

Edit: Suppose that we choose some integer $d$ and some natural number $c=c_2$. Then if we plug those values into $$ c_{n+1}=\frac{c_n(c_n+n+d)}n $$ and observe the behavior of this recursively ...
27
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0answers
702 views

Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?

In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail. QUESTION Is there any (added: ...
6
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2answers
396 views

Does this sequence of ratios of digit sums have a limit?

I asked this question a few hours ago on MathStackExchange and there it received some attention but we still do not have a proof so I decided to ask it here also, in an unchanged form, and here it is: ...
4
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0answers
142 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 ...
1
vote
1answer
70 views

Reference request for multiple free sequences

Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or primitive sequence. For example it is easy to see that $\...
3
votes
1answer
77 views

Source coding lexicographic index of finite alphabet sequence with weight (partitions)

My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint: $$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum_{j=1}^n x_j =...
4
votes
1answer
235 views

Another integral that has a closed form involving finite series of $\zeta(2k+1)$'s. Could it be reflexive?

In the context of a series of questions here, here and here, about closed form expressions involving finite series of $\zeta(2k+1)$'s for certain integrals, I would like to raise another one: $$f(n):=...
2
votes
0answers
58 views

Do almost all zeros of linear recurrence come from scaling or cancellation?

Let $a(n)$ be linear recurrence with constant coefficient of order $t$. Assume $a(n)=\sum_{i=0}^t c_i r_i^n$ where $r_i$ are the roots of the companion polynomial and $c_i$ are algebraic numbers. ...
1
vote
0answers
98 views

In search of multiple expressions for a sequence

The sequence $a_n=\sum_{k=0}^n\binom{n}k^24^k$ is listed on OEIS along with a couple of combinatorial interpretations. What interested me at the moment is the plethora of binomial single-sums for the ...
2
votes
1answer
119 views

What are non-trivial facts about the sequence of averages of digits of an integer sequence?

Write $A_{10}(k)$ for the average of the base-10 digits of a positive integer $k$: $A_{10}(k):=\tfrac{1}{L+1}(d_0+\dots+d_L)$, where $k=\sum_{i=0}^L d_i 10^i$ with $d_i\in\{0,\dots,9\}$ I wonder if ...
3
votes
2answers
149 views

Determining the asymptotic behavior of a sequence

I've encountered the following sequences $$ a_k=2^{k+1}\sum_{j=0}^{k-1}a_{k-1-j}a_j,\;a_0=1 $$ $$ b_k=(k+1)\sum_{j=0}^{k-1}b_{k-1-j}b_j,\;b_0=1. $$ I would like to have an estimate of the growth of ...