mscroggs.co.uk
mscroggs.co.uk

subscribe

Blog

 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 "number" in the box below (case sensitive):

Archive

Show me a random blog post
 2020 

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

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

Archive

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