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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_{...
ntorai's user avatar
  • 21
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}...
Iosif Pinelis's user avatar
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, ...
Mrexcel's user avatar
  • 31
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-...
GH from MO's user avatar
  • 99.1k
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 ...
1001's user avatar
  • 726
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}}...
Max Alekseyev's user avatar
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 ...
Tong Lingling's user avatar
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 ...
Max Alekseyev's user avatar
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 $p$" ...
Joachim König's user avatar
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",...
Max Alekseyev's user avatar
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 $\mathbb{Q}(\sqrt{6})$ ...
Stanley Yao Xiao's user avatar

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