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Matching sequences of 0’s and 1’s and differences of Lucas numbers

November 6, 2017 By Gary Ernest Davis

Jim Tanton (@jamestanton) asks:

“A row of 0s & 1s repeats in blocks of 4, another below it repeats in blocks of 5. Count how many 0s line up over 20 terms. What notice?”

Curious question: what could we be looking for, or hoping to notice?

We could, of course, try all 2^4=16 rows of 0’s and 1’s of length 4, together with all 2^5=32 rows of 0’s and 1’s of length 5 – giving us 16\times 32 = 512 possibilities – and check in each case the times the 0’s line up. And maybe we should do just that.

For now let’s just generate a random example and see what we might notice.

Here’s some Mathematica code to construct a random example:

A = Table[RandomInteger[{0, 1}], 4]
AA = Flatten[Table[A, 5]]
B = Table[RandomInteger[{0, 1}], 5]
BB = Flatten[Table[B, 4]]
Apositions = Flatten[Position[AA, 0]]
Bpositions = Flatten[Position[BB, 0]]
common=Intersection[Apositions, Bpositions];

Print[“There are  “,Length[common], ” positions where the 0’s line up”]

and running this once gave the following:

{0, 0, 0, 1}

{0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1}

{1, 0, 0, 0, 0}

{1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0}

{1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 18, 19}

{2, 3, 4, 5, 7, 8, 9, 10, 12, 13, 14, 15, 17, 18, 19, 20}

There are  12 positions where the 0’s line up

Nothing pops out, at least not to me.

Doing this a few more times doesn’t indicate any pattern of the number of times 0’s line up.

So let’s do this a bunch of times, say 1,000,000 times.

What numbers of line ups do we get?

Here’s some Mathematica code to do that:

n = 1;
L = {};
While[n <= 1000000,
A = Table[RandomInteger[{0, 1}], 4];
AA = Flatten[Table[A, 5]];
B = Table[RandomInteger[{0, 1}], 5];
BB = Flatten[Table[B, 4]];
AAA = Flatten[Position[AA, 0]];
BBB = Flatten[Position[BB, 0]];
L = {L, Length[Intersection[AAA, BBB]]}; n++]
L = Union[Flatten[L]]

and here’s the result:

{0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 20}

The numbers that are missing from this list are 7, 11, 13, 14, 17, 18, 19

Curiously, this number sequence – 7, 11, 13, 14, 17, 18, 19 – occurs in the Online Encyclopedia of Integer Sequences as part of a larger sequence: namely the ordered sequence of nonnegative differences f-4g, where f and g are Lucas numbers.

Is the sequence 7, 11, 13, 14, 17, 18, 19 exactly the list of forbidden number of times the 0’s line up: we can check this by examining all possible sequences of 0’1 and 1’s of length 4 and 5.

Why is there a connection with Lucas numbers? Is this a coincidence, or is there a story here?

 

 

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