Recamán's sequence
a₀ = 0. For each step n, aₙ is aₙ₋₁ − n when that integer is positive and not already in the sequence. Otherwise aₙ is aₙ₋₁ + n. The first terms are 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24. PictorX draws each hop as a semicircle, lists the reason for the choice, and marks the smallest positive integer the prefix has not reached. Semicircles alternate above and below the number line. Drag to move, scroll to zoom, and double-click to fit. Turn on Sound to hear each hop.
How it works
- Open this page and watch the semicircles grow above and below the number line.
- Switch to Steps to read why each term was chosen, or to Missing numbers to see the smallest gap.
- Turn on Sound, then save a PNG or continue in the PictorX editor.
Backward when the spot is free
The sequence starts at 0. Step n subtracts n when the result is a positive integer that has not appeared yet. If that spot is negative or already used, the step adds n instead. Zero is the starting term, so a hop back onto 0 is refused. The first fifteen terms are 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24.
Semicircles, then the gaps
A semicircle of diameter n joins each term to the next. Odd steps arch above the line and even steps arch below. Teal means the step hopped backward into a free integer. Amber means it had to jump forward. After the first fifteen terms, 4 is still missing. It appears at step 131. Missing numbers names the smallest gap in whatever prefix you draw.
A conjecture, still open
Neil Sloane conjectured that every nonnegative integer eventually appears. The conjecture is unproved. Benjamin Chaffin reported in February 2026 that after 10^612 terms, 852,655 is still the smallest missing number. The drawing, the step table, and the melody all run in your browser.
Frequently asked questions
What is Recamán's sequence?
The sequence with a₀ = 0 and, for n > 0, aₙ = aₙ₋₁ − n when that value is positive and new. Otherwise aₙ = aₙ₋₁ + n. It begins 0, 1, 3, 6, 2, 7, 13, 20, 12, 21, 11, 22, 10, 23, 9, 24.
Why does step 7 land on 20?
13 − 7 = 6, and 6 is already in the sequence, so the backward hop is refused. The step adds 7 and lands on 20.
What do the semicircles show?
Each arc connects one term to the next and has diameter n. Odd steps go above the number line and even steps go below. Teal arcs hopped backward. Amber arcs jumped forward.
Does every number appear?
It is an open conjecture, due to Neil Sloane, that every nonnegative integer appears. After 10^612 terms, 852,655 is still the smallest missing number.
Does the math run in my browser?
Yes. The sequence, the arcs, the gaps, and the melody are computed on your device. Save a PNG or open the canvas in the PictorX editor.
Related tools
Recamán's sequence
How Recamán's sequence hops backward by n when that spot is free, why it sometimes jumps forward, and why 852,655 is still missing after 10^612 terms.
The Fibonacci sequence
Each Fibonacci number is the sum of the two before it. Watch the squares spiral, the ratios settle on the golden ratio, and Binet's formula rebuild the same numbers.
The Collatz Conjecture
Explore the Collatz 3n+1 problem in your browser. Walk hailstone sequences, grow a geometric Collatz tree, and zoom the continuous Collatz fractal.
Ulam spiral
Draw the Ulam spiral in your browser. Mark the primes, read the quadratic on each diagonal, and open Euler's unbroken run of 40 primes.
Multiplication Circles
Join each number on a circle to its multiple. Draw a times table, a cardioid, or a string figure on paper, chalk, or wood.
Langton's Ant
Run Langton's Ant in your browser. Watch symmetry, chaos and the 104-step highway, then try multi-colour rules, several ants, a hex grid and turmites.
Ready to try it?
Open PictorX in your browser — no install, no credit card required to start.
Go to Editor