mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

Proving a conjecture

 2019-06-19 
Last night at MathsJam, Peter Kagey showed me a conjecture about OEIS sequence A308092.
A308092
The sum of the first \(n\) terms of the sequence is the concatenation of the first \(n\) bits of the sequence read as binary, with \(a(1) = 1\).
1, 2, 3, 7, 14, 28, 56, 112, 224, 448, 896, 1791, 3583, 7166, ...
To understand this definition, let's look at the first few terms of this sequence written in binary:
1, 10, 11, 111, 1110, 11100, 111000, 1110000, 11100000, 111000000, ...
By "the concatenation of the first \(n\) bits of the sequence", it means the first \(n\) binary digits of the whole sequence written in order: 1, then 11, then 110, then 1101, then 11011, then 110111, and so on. So the definition means:
As we know that the sum of the first \(n-1\) terms is the first \(n-1\) digits, we can calculate the third term of this sequence onwards using: "\(a(n)\) is the concatenation of the first \(n\) bits of the sequence subtract concatenation of the first \(n-1\) bits of the sequence":

The conjecture

Peter's conjecture is that the number of 1s in each term is greater than or equal to the number of 1s in the previous term.
I'm going to prove this conjecture. If you'd like to have a try first, stop reading now and come back when you're ready for spoilers. (If you'd like a hint, read the next section then pause again.)

Adding a digit

The third term of the sequence onwards can be calculated by subtracting the first \(n-1\) digits from the first \(n\) digits. If the first \(n-1\) digits form a binary number \(x\), then the first \(n\) digits will be \(2x+d\), where \(d\) is the \(n\)th digit (because moving all the digits to the left one place in binary is multiplying by two).
Therefore the different is \(2x+d-x=x+d\), and so we can work out the \(n\)th term of the sequence by adding the \(n\)th digit in the sequence to the first \(n-1\) digits. (Hat tip to Martin Harris, who spotted this first.)

Carrying

Adding 1 to a binary number the ends in 1 will cause 1 to carry over to the left. This carrying will continue until the 1 is carried into a position containing 0, and after this all the digits to the left of this 0 will remain unchanged.
Therefore adding a digit to the first \(n-1\) digits can only change the digits from the rightmost 0 onwards.

Endings

We can therefore disregard all the digits before the rightmost 0, and look at how the \(n\)th term compares to the \((n-1)\)th term. There are 5 ways in which the first \(n\) digits could end:
We now look at each of these in turn and show that the \(n\)th term will contain at least as many ones at the \((n-1)\)th term.

Case 1: \(00\)

If the first \(n\) digits of the sequence are \(x00\) (a binary number \(x\) followed by two zeros), then the \((n-1)\)th term of the sequence is \(x+0=x\), and the \(n\)th term of the sequence is \(x0+0=x0\). Both \(x\) and \(x0\) contain the same number of ones.

Case 2: \(010\)

If the first \(n\) digits of the sequence are \(x010\), then the \((n-1)\)th term of the sequence is \(x0+1=x1\), and the \(n\)th term of the sequence is \(x01+0=x01\). Both \(x1\) and \(x01\) contain the same number of ones.

Case 3: \(01...10\)

If the first \(n\) digits of the sequence are \(x01...10\), then the \((n-1)\)th term of the sequence is \(x01...1+1=x10...0\), and the \(n\)th term of the sequence is \(x01...10+1=x01...1\). \(x01...1\) contains more ones than \(x10...0\).

Case 4: \(01\)

If the first \(n\) digits of the sequence are \(x01\), then the \((n-1)\)th term of the sequence is \(x+0=x\), and the \(n\)th term of the sequence is \(x0+1=x1\). \(x1\) contains one more one than \(x\).

Case 5: \(01...1\)

If the first \(n\) digits of the sequence are \(x01...1\), then the \((n-1)\)th term of the sequence is \(x01...1+1=x10...0\), and the \(n\)th term of the sequence is \(x01...1+1=x10...0\). Both these contain the same number of ones.

In all five cases, the \(n\)th term contains more ones or an equal number of ones to the \((n-1)\)th term, and so the conjecture is true.

Similar posts

Mathsteroids
Building MENACEs for other games
Big Ben Strikes Again
Christmas (2016) is over

Comments

Comments in green were written by me. Comments in blue were not written by me.
 Add a Comment 


I will only use your email address to reply to your comment (if a reply is needed).

Allowed HTML tags: <br> <a> <small> <b> <i> <s> <sup> <sub> <u> <spoiler> <ul> <ol> <li>
To prove you are not a spam bot, please type "odd" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

Jul 2020

Happy e3√3-13 Approximation Day!

May 2020

A surprising fact about quadrilaterals
Interesting tautologies

Mar 2020

Log-scaled axes

Feb 2020

PhD thesis, chapter ∞
PhD thesis, chapter 5
PhD thesis, chapter 4
PhD thesis, chapter 3
Inverting a matrix
PhD thesis, chapter 2

Jan 2020

PhD thesis, chapter 1
Gaussian elimination
Matrix multiplication
Christmas (2019) is over
 2019 
▼ show ▼
 2018 
▼ show ▼
 2017 
▼ show ▼
 2016 
▼ show ▼
 2015 
▼ show ▼
 2014 
▼ show ▼
 2013 
▼ show ▼
 2012 
▼ show ▼

Tags

determinants bubble bobble estimation coins royal institution rhombicuboctahedron draughts weak imposition news probability simultaneous equations big internet math-off computational complexity palindromes pizza cutting plastic ratio triangles javascript twitter manchester world cup preconditioning weather station reuleaux polygons wave scattering pi approximation day hats mathsteroids stickers royal baby matrix of cofactors folding paper london sorting rugby matrix multiplication european cup polynomials braiding golden ratio chebyshev curvature puzzles platonic solids data visualisation latex interpolation accuracy radio 4 php light manchester science festival matrices quadrilaterals electromagnetic field talking maths in public inline code mathsjam london underground golden spiral numerical analysis final fantasy cross stitch fractals exponential growth martin gardner mathslogicbot arithmetic nine men's morris dates frobel python error bars convergence menace advent calendar football countdown finite element method sport bodmas matrix of minors hexapawn chess raspberry pi geometry binary signorini conditions logs matt parker tennis squares gerry anderson misleading statistics harriss spiral pythagoras game show probability christmas card a gamut of games video games craft ternary game of life phd logic propositional calculus books geogebra graph theory trigonometry captain scarlet reddit bempp approximation statistics sound flexagons machine learning inverse matrices christmas asteroids dragon curves programming graphs speed dataset go map projections noughts and crosses people maths tmip national lottery games wool gaussian elimination chalkdust magazine ucl pi pac-man sobolev spaces boundary element methods hannah fry cambridge the aperiodical realhats oeis data folding tube maps

Archive

Show me a random blog post
▼ show ▼
© Matthew Scroggs 2012–2020