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Tag Archives: integers
A combinatorial proof: PIE a la mode!
Continuing from my last post in this series, we’re trying to show that , where is defined as which is what we get when we start with a sequence of consecutive th powers and repeatedly take successive differences. Recall that … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
Comments Off on A combinatorial proof: PIE a la mode!
A combinatorial proof: counting bad functions
In a previous post we derived the following expression: . We are trying to show that , in order to show that starting with a sequence of consecutive th powers and repeatedly taking successive differences will always result in . … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
1 Comment
A combinatorial proof: functions and matchings
We’re trying to prove the following equality (see my previous post for a recap of the story so far): In particular we’re trying to show that the two sides of this equation correspond to two different ways to count the … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, function, integers, matching, powers
5 Comments
A combinatorial proof: the story so far
In my last post I reintroduced this seemingly odd phenomenon: Start with consecutive integers and raise them all to the th power. Then repeatedly take pairwise differences (i.e. subtract the first from the second, and the second from the third, … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, integers, powers
1 Comment
A combinatorial proof: reboot!
More than seven years ago I wrote about a curious phenomenon, which I found out about from Patrick Vennebush: if you start with a sequence of consecutive th powers, and repeatedly take pairwise differences, you always end up with , … Continue reading
Posted in arithmetic, combinatorics, proof
Tagged consecutive, difference, integers, powers
11 Comments
The Recamán sequence
I recently learned about a really interesting sequence of integers, called the Recamán sequence (it’s sequence A005132 in the Online Encyclopedia of Integer Sequences). It is very simple to define, but the resulting complexity shows how powerful selfreference is (for … Continue reading
Posted in arithmetic, recursion, sequences
Tagged difference, integers, Recamán, repeat, sequence
5 Comments
Differences of powers of consecutive integers, part II
If you spent some time playing around with the procedure from Differences of powers of consecutive integers (namely, raise consecutive integers to the th power, and repeatedly take pairwise differences until reaching a single number) you probably noticed the curious … Continue reading
Posted in arithmetic, iteration, pascal's triangle
Tagged binomial coefficients, consecutive, difference, integers, powers
3 Comments
Differences of powers of consecutive integers
Patrick Vennebush of Math Jokes 4 Mathy Folks recently wrote about the following procedure that yields surprising results. Choose some positive integer . Now, starting with consecutive integers, raise each integer to the th power. Then take pairwise differences by … Continue reading
Posted in arithmetic, pattern
Tagged consecutive, difference, integers, powers, surprising
16 Comments
More fun with infinite decadic numbers
This is the sixth in a series of posts on the decadic numbers (previous posts: A curiosity, An invitation to a funny number system, What does “close to” mean?, The decadic metric, Infinite decadic numbers). Last time I left you … Continue reading
Posted in arithmetic, infinity, number theory
Tagged decadic, decimal, fractions, integers, representation
4 Comments
The decadic metric
Continuing my series of posts exploring the decadic numbers… in my previous post, I explained that we will define a new “size function”, or metric, different from the usual “absolute value”, and written . Two numbers will be “close to” … Continue reading
Posted in arithmetic, number theory, pattern
Tagged decadic, distance, integers, metric, number line, numbers, padic, soup
5 Comments