1
vote
1answer
31 views

How to convert linear recurrence to a tiling question

If I have some linear recurrence of form $$f(n) = a_1f(n-1) + a_2f(n-2) + a_3f(n-3) + \cdots + a_kf(n-k)$$ How does this translate to tilings? For example the Fibonacci sequence is the same as ...
3
votes
1answer
50 views

2D Pattern in recursive digit sum of consecutive x^n

Take the recursive decimal digit sum of consecutive binary numbers $2^n$ as $n \to \infty$. You'll see something like this: n 2^n (recursive) sum of digits ...
0
votes
1answer
45 views

How often does $p^k$ divide the Fibonacci numbers?

I would like to know about the Fibonacci numbers $F_n = 1,1,2,3,5,8, \dots$ in $\mathbb{Z}/p^k\mathbb{Z}$. $$ \mathbb{P}[p^k \text{ divides } F_n ] = \frac{\#\{1 \leq n\leq N: F_n \equiv 0 \mod ...
1
vote
0answers
36 views

Infinitely many primes in second-order recurrence

I just wondered about the following question: Suppose that we are given a homogeneous second-order recurrence relation, $x_{n+2}+ax_{n+1}+bx_n=0$ for all $n\in\mathbb{N}$. Can we choose integers ...
1
vote
2answers
66 views

number theory of coefficients in an infinite sequence of polynomials

EDIT: equivalent formulation by Hurkyl in comments: if $n$ is odd and $p^\nu \parallel n$ and $n > 2k,$ then $$ p^{(\nu + 2 + 2 k - n)} \; | \; \sum_j \left( \begin{array}{c} n \\ 2j \end{array} ...
2
votes
1answer
65 views

Define a recursive sequence for the following formula f(n) = n(n+1)

Define a recursive sequence for the following formula f(n) = n(n+1). Preferably one only defined by previous $a_n$ terms, i.e., no 'n' terms. If possible that is. So for example the following ...
1
vote
0answers
60 views

Solving a Recurrence Relation With Summation and Tau Function

How can I solve the following: $$ T(n) = \sum_{i = 1}^{d(n) - 2}T(v_i) + \sum_{i = d(n) - 1}^{n - 1}c + c' $$ Where $d(n)$ is the Tau function, and v is the set of values dividing n. e.g. $d(18) = ...
-1
votes
2answers
115 views

how to find nth term of different fibonacci series with golden ratio [duplicate]

what i know : if i want to find $Nth$ term of a fibonacci series like : 1 1 2 3 5 8 13 21 ....... then to find $6th$ term we use golden ratio ...
2
votes
0answers
92 views

How to solve a ceiling expression or recurrence equation?

I am trying to express the following in term of n, $$\underbrace{\hbox{$ \left \lceil \frac{3}{2}.\left \lceil { \frac{3}{2}.\left \lceil { \ldots \left \lceil \frac{3}{2}.\left \lceil { ...
11
votes
1answer
184 views

Family of GCDs all equal to $2$

Is it true that $$\gcd\left(5^{2^n} + 1, 13^{2^n} + 1\right) = 2$$ for all $n \in \mathbf{Z}_{\geq 0}$? I'm continually stumped with this and verifying it numerically is quite expensive very ...
2
votes
2answers
85 views

Does there exist an infinite sequence $p_0,p_1,p_2…$ of prime numbers such that $p_k=4 p_{k-1}\pm 1$

$k \in Z^+$ firstly we know that there exists infinetly many primes of the form $4n+1$ by FTA also we see that if we consider finite primes say to $n$ then the recursive formular can be expressed ...
3
votes
0answers
44 views

Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
1
vote
1answer
29 views

Recurrence relation related proof

Find a recurrence relation for the number of ternary string that do not contain 00 or 11 .
0
votes
0answers
34 views

Is crossdirectional partial tetration of order $n^c$?

The following animation shows a sum of a matrix where the parameter $c=0$ gives a straight line in pink, and as $c \rightarrow 1$ it approaches the Chebyshev $\psi$ function, the blue staircase. This ...
3
votes
2answers
60 views

Does $E^2 \; ( E \approx 1.2640847\ldots)$ equal $D \approx 1.5979102\ldots$?

Does $E^2=D$? Where $E$ is a constant used in the closed form of the Sylvester Sequence (see: Closed form formula and asymptotics) and $D$ is a constant for the closed formula of the sequence A007018 ...
0
votes
1answer
80 views

Fractional-Recursive Sequence

Here from the fraction set we have a really hard question to be answered...suppose that a sequence is defined as a(n) = a(n-1) - 1/a(n-1), where a(0) is given. ...you already know what I'm asking you ...
0
votes
1answer
46 views

Solve non-linear recurrence

I am trying to solve the following recurrence (i.e. determine $a_n$). I'm not entirely sure how to proceed, I have only encountered linear ones so far. This is not a homework, I'm just doing it for ...
9
votes
3answers
276 views

The integer $c_n$ in $(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write $$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$ where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove or disprove ...
1
vote
0answers
66 views

Deriving this recursive expression for Riemann Prime Counting Function?

Why does this work? $f(n,k,1)=0$ $f(n,k,j)= \frac{1}{k} - f(\lfloor\frac{n}{j}\rfloor, k+1, \lfloor\frac{n}{j}\rfloor) + f(n,k,j-1)$ Here, f(n,1,n) computes the Riemann Prime Counting Function. ...
1
vote
2answers
921 views

Need some help with this recurrence equation

I'm self studying from a book I bought to learn more about algorithms and I've been trying most of the exercises in that book, so this is not a homework. Anyways, the relation I'm trying to solve is ...
1
vote
1answer
120 views

Alternative solutions to $n^2+n = k^2+k + 2kn$

Consider this equation: $n^2+n = k^2+k + 2kn$ I want to find the set of non-negative integer n,k that satisfies the equation. I tried to write $n$ as $k$ by solving the equation with $n$ as root ...
9
votes
2answers
97 views

Proving that $a_n$ is an integer for every $n$

For every $k\ge1$ integer number if we define the sequence : $a_1,a_2,a_3,...,$ in the form of :$$a_1=2$$ $$a_{n+1}=ka_n+\sqrt{(k^2-1)(a^2_n-4)}$$ For every $n=1,2,3,....$ how to prove that $a_n$ is ...
2
votes
2answers
85 views

Twice a triangle is triangle

The question is to prove that there are infinitely many triangular numbers $T_n$ where $2 \times T_n$ is also a triangular number, and give the first few as an example. My attempt: $$2 \cdot {x(x+1) ...
4
votes
1answer
77 views

$ n $ lines intersections

As we all know, $ n $ lines which are not coincident may have some intersection points in an Euclid plane. And we define the set of the number of intersection points $ n $ lines can form is $ ...
3
votes
5answers
714 views

How to create a generating function / closed form from this recurrence?

Let $f_n$ = $f_{n-1} + n + 6$ where $f_0 = 0$. I know $f_n = \frac{n^2+13n}{2}$ but I want to pretend I don't know this. How do I correctly turn this into a generating function / derive the closed ...
2
votes
1answer
109 views

Show that Fermat number $F_n$ and its index $n$ are coprime.

I want to show that $\gcd(F_n,n)=1$, where $F_n=2^{2^n}+1$. How to prove this? I can show that that $\gcd(F_n, F_m)=1$ for any natural $n$ and $m$, and that $F_{n+1}=(F_n)^2-2F_n+2=F_0\dots ...
2
votes
2answers
210 views

How do you find the closed form of these recurrence relations?

I found these two recurrence relations in an old textbook and was hoping someone could show me how to solve them for their closed form. If not, a final answer would also be appreciated, as it helps ...
4
votes
2answers
123 views

Does this problem have a name?

Recently our lecturer told us that it is an unsolved mathematical problem if the following while loop aka iteration ever terminates. Unfortunately I forgot to ask him what it is called. If someone ...
2
votes
6answers
275 views

I know that, $S_{2n}+4S_{n}=n(2n+1)^2$. Is there a way to find $S_{2n}$ or $S_{n}$ by some mathematical process with just this one expression?

$S_{2n}+4S_{n}=n(2n+1)^2$, where $S_{2n}$ is the Sum of the squares of the first $2n$ natural numbers, $S_{n}$ is the Sum of the squares of the first $n$ natural numbers. when, $n=2$ ...
11
votes
1answer
290 views

Why do the Fibonacci numbers recycle these formulas?

The Fibonacci numbers $F_n = 0, 1, 1, 2, 3, 5, 8, 13, \dots$ obey the following recurrence relations, $ \begin{aligned} &F_{n}-\;F_{n-1}-F_{n-2} = 0\\[1.5mm] &F_{n-1}^3-F_{n}^3-F_{n+1}^3 = ...
17
votes
2answers
857 views

Self-avoiding walk on $\mathbb{Z}$

How many sequences $a_1,a_2,a_3,\dotsc$, satisfy: i) $a_1=0$ ii) ($a_{n+1}=a_n-n$ or $a_{n+1}=a_n+n$) iii) $a_i\neq a_j$ for $i\neq j$ iiii) $\mathbb{Z}=\{a_i\}_{i>0}$ Are the two alternating ...
2
votes
0answers
254 views

Bell-like recurrence

Let $$A(n)=\sum_{k=0}^{n-1}\binom{n}{k}A(k)+n!,\quad A(0)=1$$ $$B(n)=\sum_{k=0}^{n-1}\binom{n}{k}B(k)-n!-n!\sum_{k=1}^{n}\frac{1}{k!},\quad B(0)=-1.$$ I'm interested in computing $S(n)=A(n)+B(n)$ ...
3
votes
0answers
57 views

Properties preserved under the “reversal” of a recurrence equation

Consider the recurrence equation $u_n = f(u_{n-1},\ldots, u_{n-k})\;,$ defined for $n=0,1,2\ldots$. If this is a linear recurrence and the coefficient function on $u_{n-k}$ is nonzero in ...
4
votes
0answers
174 views

Is this a recurrence for the Mertens function plus 2?

If we define a symmetric array: $$T(1,1)=3,\; T(1,2)=2,\; T(2,1)=2$$ $$T(1,k)=\frac{-T(n,k-1)-\sum\limits_{i=2}^{k-1}T(i,k)}{k+1}+T(n,k-1)$$ $$ ...
8
votes
1answer
127 views

Prove that a holonomic (p-recursive) difference equation returns only integral values

Consider the recurrence given by $(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$ $a_0 = 1, a_1 = 3$. Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral ...
6
votes
1answer
257 views

Common terms in general Fibonacci sequences

Mathworld notes that "The Fibonacci and Lucas numbers have no common terms except 1 and 3," where the Fibonacci and Lucas numbers are defined by the recurrence relation $a_n=a_{n-1}+a_{n-2}$. For ...
7
votes
1answer
179 views

Diagonal of the double sequence $(n+1)v_{h,n+1}-(2h+1)v_{h,n}-nv_{h,n-1}=0$

Update: it is not possible to reply to this question without additional information. My comment below: "I have to agree with you that one "cannot derive (2) from (1) alone". Now it seems to me that ...
3
votes
2answers
213 views

Recurrence relation for $a_{n}$ where $6a_{n}$ and $10a_{n}$ are both triangular

According A180926, the elements of the set {$a:\exists m,n|60a=5n^2+5n=3m^2+3m$} satisfy the following recurrence relation: $$a_{n}=\frac{62a_{n-1}+1+\sqrt{(48a_{n-1}+1)(80a_{n-1}+1)}}{2}$$ ...