September 24, 2020

# The succession of Padovan | Science

The numbers 187 and 2019 can be decomposed into sums of squares and cubes in different ways, as it was said last week; Here are some of the shorter ones, found by our usual commentators Oli and Theram:

187 = 13² + 3² + 3²;

2019 = 43² + 13² + 1²

187 = 5³ + 3³ + 3³ + 2³

2019 = 11³ + 7³ + 7³ + 1³ + 1³

The sequence 16, 1, 2, 3, 5, 1, 1 … is an example of those phenomena or events that go unnoticed by their very "normality", like those noises of which we are only aware when they cease. In many houses there are still wall clocks that give the rooms and the hours, and in almost all the houses one of those clocks comes in once a year to give the New Year bells. At the end of one day and start another, those clocks give 16 chimes: 4 of the rooms plus 12 of the hour; a quarter of an hour later, 1 bell announces the first quarter of the following hour … And the fifth term of the sequence, 5, corresponds to the four quarters of the one plus the bell of the hour. As a clue, I said that the sequence was inspired by a very recent and very "sounded" event (the New Year's chimes); but … There arises a question that I pose as a meta-construct: is this sequence acceptable as a logical-mathematical conundrum, or is it one of those trap-questions that when they tell you the answer you exclaim in outrage: "That's not worth it!".

A few years ago, a malevolent child drove me crazy by showing me the following number sequence: 2, 5, 5, 4, 5, 6, 3 … What number does it follow? It took me several days to find the solution, and by chance, and my first reaction was to exclaim mentally: "Damn kid!" But then I thought that this kind of atypical sequences, or slightly cheaters, they invite us to exercise lateral thinking, so welcome. One clue: this sequence has a lot to do with the chimes. And a metapista: the previous track can be deceiving.

### A family succession

Since we are talking about numerical sequences and we recently talked about Ian Stewart, it is inevitable to mention Padavan's succession:

1, 1, 1, 2, 2, 3, 4, 5, 7 …

What number does it follow? (It is easy to find the answer to this and other questions related to the succession of Padovan in the network, but do not deprive yourself of the pleasure of finding them without help).

The succession was popularized in 1996 by Ian Stewart in his section of mathematical pastimes of Scientific American, but it is the work of British mathematician and architect Richard Padovan, hence its name.

Surely more than one, although he did not know this numerical sequence, will have thought to see it: "I find it familiar". With what other famous succession is related? And for the most astute readers: what does it have to do with the equation x3 – x – 1 = 0?

Carlo Frabetti He is a writer and mathematician, a member of the New York Academy of Sciences. He has published more than 50 scientific dissemination works for adults, children and young people, among them Damn physics, Damn mathematics or The big game. He was a screenwriter The Cristal ball.