Super Safi’s Monday Morning Math Mayhem 27 – Padovan Sequence

Morning Mathematical Monsters & Maniacs!

(Today’s post is sponsored by the letter “M”)

 

Hi, I’m Super Safi and you may remember me from such stats and strategy posts as Kwik-E-Mart Farming and the advanced losing-to-win Superheroes battle strategy.

Over the past 600+ episodes, The Simpsons has taken us on an amazing mathematical journey involving fractions, probability, Fermat’s last theorem, and hundreds of other aspects from the wonderful world off mathematics.

And what better way to start your week, then by discussing math Monday morning?

101 years ago today, the “eleventh hour of the eleventh day of the eleventh month” of 1918, the armistice took effect marking a ceasefire between the allied forces and Germany, leading to the end of World War I. In honour of the signing of the armistice, we are celebrating Rememberance Day in Canada, Veterans Day in the US, and similar holidays in countries around the world.

This week we look at the first mention of World War I and armistice in The Simpsons.

 

In Bart the General (Season 01, Episode 05), after defending Lisa from school bully Nelson, Bart becomes Nelson’s latest school bullying target. Having become sick of the harassment and torment, Bart, Grampa Simpson, and Herman rally the town’s children into fighting back against Nelson and his cronies.

BART: “Sound off!”

CHILDREN: “One, two!”

BART: “Sound off!”

CHILDREN: “Three, four!”

MILHOUSE: “Nelson’s at the Elm Street video arcade.”

BART: “Intelligence indicates he shakes down kids for quarters at the arcade, then heads to the Kwik-E-Mart for a cherry Squishee.”

HERMAN: “And that’s where we’ll hit him. When he leaves the Kwik-E-Mart, we start the saturation bombing. Got the water balloons?”

BART: “200 rounds, sir. (holds up a water balloon) Is it OK if they say “Happy Birthday” on the side?”

HERMAN: “I’d rather they say “Death From Above”, but I guess we’re stuck. OK.”

A lot of numbers thrown around so far. Season 1, episode 5, sound off has 1, 2, 3, 4, and they’ve prepared 200 water balloons. So today we look at a sequence that contains all of these numbers.

 

Padovan Sequence

The Padovan sequence is the sequence of integers P(x) defined by the initial values:

P(0)   =   1

P(1)   =   1

P(2)   =   1

This is followed by the recurrence:

P(x)   =   P(x-2)   +   P(x-3)

 

Plugging in values for the recurrence, we can see the first few numbers in the Padovan sequence are:

P(0)   =   1

P(1)   =   1

P(2)   =   1

P(3)   =   P(1)   +   P(0)   =   1   +   1   =   2

P(4)   =   P(2)   +   P(1)   =   1   +   1   =   2

P(5)   =   P(3)   +   P(2)   =   2   +   1   =   3

P(6)   =   P(4)   +   P(3)   =   2   +   2   =   4

P(7)   =   P(5)   +   P(4)   =   3   +   2   =   5

P(8)   =   P(6)   +   P(5)   =   4   +   3   =   7

P(9)   =   P(7)   +   P(6)   =   5   +   4   =   9

P(10)   =   P(8)   +   P(7)   =   7   +   5   =   12

P(11)   =   P(9)   +   P(8)   =   9   +   7   =   16

P(12)   =   P(10)   +   P(9)   =   12   +   9   =   21

P(13)   =   P(11)   +   P(10)   =   16   +   12   =   28

P(14)   =   P(12)   +   P(11)   =   21   +   16   =   37

P(15)   =   P(13)   +   P(12)   =   28   +   21   =   49

P(16)   =   P(14)   +   P(13)   =   37   +   28   =   65

P(17)   =   P(15)   +   P(14)   =   49   +   37   =   86

P(18)   =   P(16)   +   P(15)   =   65   +   49   =   114

P(19)   =   P(17)   +   P(16)   =   86   +   65   =   151

P(20)   =   P(18)   +   P(17)   =   114   +   86   =   200

 

The sequence is named after architect Richard Padovan (born 1935), who came across it in 1994 while studying the works of Dutch architect Hans van der Laan (1904-1991). While Padovan acknowledges the sequence was discovered by van der Laan; in the field of mathematics, it was British mathematician, and sci-fi novelist, Ian Stewart (born 1945) who named the sequence after Padovan.

In 2004, Irish mathematician Paul Barry discovered Padovan sequence can be calculated by adding diagonals in Pascal’s Triangle:

 

Sadly, the number 6 representing the number of articles in the armistice Bart and Nelson sign, does not appear in the Padovan sequence.

HERMAN: “Armistice Treaty, article four: Nelson is never again to raise his fists in anger. Article Five: Nelson recognizes Bart’s right to exist. Article Six: Although Nelson shall have no official power he shall remain a figurehead of menace in the neighbourhood.”

 

 

Now that we’ve completed a look at Padovan sequence, why not show your love for math and sequences with your own Math Mayhem shirt or hoodie.

If you love math or enjoy reading these posts, don’t forget to stop by the Addicts Shop and check out all the paraphernalia, including the Math Mayhem shirts and hoodies.

Were you familiar with this sequence? Can you figure out the next few numbers in the sequence? Do you remember the episode from the show? Is your country commemorating a holiday today? Sound off in the comments below. You know we love hearing from you.