## New answers tagged integer-sequences

2
votes

### Negated Fibonacci and the floor function

As a complement to Iosif's answer, I'll give the inductive proof. First, some preliminaries: note that by recursively applying the defining relations,
$$
F_n=F_{n-1}+F_{n-3}+...+F_2+1=1-(F_{-(n-1)}+F_{...

2
votes

### Negated Fibonacci and the floor function

We have
$$F_{-n}=\frac{(-a_-)^n-(-a_+)^n}{\sqrt5} \tag{1}\label{1}$$
with
$$a_\pm:=\frac{1\pm\sqrt5}2.$$
We also have the easy formula for $\sum_{j=j_1}^{j_2}(p+qj)x^j$, which yields
$$\sum_{i=1}^{n-1}...

3
votes

### The largest digital sum of the square of an n-digit number

After 10 +
days calculating, I got the following sequence : {13, 31, 46, 63, 81,
97, 112, 130, 148, 162, 180, 193, 211, 229, 244, 262, 277, 297,
310, 331, 343, 360, 378, 396, 406, 423, 436, 454, 469, ...

2
votes

### Small solutions of $x^2-a^3 y^2=\pm 1$

It is better to ask one question per post. Here is the answer to Q1.
Assume that $(x,y)$ is a positive integer solution of $(1)$. Then $(x,ay)$ is a solution of $X^2-aY^2=1$. Let us restrict to $a=u^2-...

11
votes

Accepted

### Suitable closed form for the A079501

Yes!
$i$ is the first part in the composition
$j + 1$ is the number of other parts in the composition
$+1$ accounts for the case that there is only one part
Summing over $i$ and $j$, we want to ...

2
votes

Accepted

### Recursion for the Chebyshev transform of $m^n$

UPDATED. The argument below is corrected.
Apparently, under Chebyshev transform of a generating function $A(x)$ OP understands a function $B(x) := C(-x^2)A(xC(-x^2))$, where $C(x):=\frac{1-\sqrt{1-4x}}...

3
votes

### Six consecutive positive integers with certain shape

With the analysis from Stanley, Joachim and Max Alekseyev, maybe I can solve my question now.
Because two different integers with the form 4k+2 can not appear in these six numbers, when modulo 4, they ...

3
votes

### Six consecutive positive integers with certain shape

This is just to show that in sextuples of interest $2x^2 - 3y^2 = -1$, while difference $2$ is not possible.
First note that neither $|2x^2-2y^2|<6$ nor $|3x^2-3y^2|<6$ is soluble in distinct ...

11
votes

### Six consecutive positive integers with certain shape

I'd like to add something from the viewpoint of heuristics.
The most restrictive condition in this problem seems to be "occurence of a number of $p$-adic valuation $1$ for a small prime $p$" ...

3
votes

Accepted

### Ask for a generating function or an explicit expression of a triangle of positive integers

The generating function:
$${\cal C}(x,y) = \sum_{n,k\geq 0} C_{n,k} x^n y^{2k}$$
has the following explicit form:
$${\cal C}(x,y) = \frac{\arctan(y)}{y(1-x(1+y^2))}.$$
For "one more problem",...

9
votes

### Six consecutive positive integers with certain shape

Given your question, it seems that one should study the pair of equations
$\displaystyle 2x^2 - 3y^2 = -1 \quad \text{and} \quad 2x^2 - 3y^2 = 2.$
(In fact, the quadratic field $\mathbb{Q}(\sqrt{6})$ ...

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