Recamán sequence

Definitions

Recamán's sequence (or Recaman's sequence): a(0) = 0; for n > 0, a(n) = a(n-1) - n if positive and not already in the sequence, otherwise a(n) = a(n-1) + n.

-- OEIS

The sequence for e.g. first 10 iterations is: 1, 3, 6, 2, 7, 13, 20, 12, 21, 11

Useful links:

• Popular Numberphile video
• Wolfram
• p5.js implementation as an online coding challenge

Demo

https://robinmackenzie.github.io/recaman/

Here is an example with 66 iterations of the sequence with primes highlighted. The prime highlight colors an arc red where the destination index is prime.

Here is an example with 400 iterations of the sequence using D3's interpolateWarm scale. The visualisation has been panned and zoomed to show detail around indices 30-150.

Javascript implementation

function recaman(iterations) {
  let sequence = [];
  let index = 0;
  for (let n=1; n<=iterations; n++) {
    sequence.push(index);
    let candidate = index - n;
    (candidate >= 0 & sequence.indexOf(candidate) < 0) ? index = candidate : index += n;
  }
  return sequence;
}

Why do a visualisation?

The Numberphile video is a good summary of this question. The sequence exhibits, in the words of Alex Bellos:

"... an interesting clash between order and chaos... "

This visualisation

The implementation builds on a number of ideas arising from the Numberphile video:

• Let the user 'play' with the graph with pan and zoom
• Let the user decide the number of iterations to visualise
• Add some colour to the visualisation as an aid to interpretation
• Per the comments on the youtube video on the p5.js implementation, show where an arc hits a prime number

To do features:

• Show where the arc hits a number from another sequence e.g. Fibonacci
• Colorise based on something other than iteration index
• Highlight an arc and see the generating items of the sequence
• Investigate 'un-fixing' the x-axis when panning on y-axis