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 
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?
