A308629 Lengths between nodes on a square spiral, such that the n-th node is closer to the (n-2)-th node than to the (n-1)-th node.

1, 2, 4, 4, 5, 6, 8, 6, 10, 10, 9, 12, 13, 11, 15, 15, 15, 17, 18, 17, 20, 21, 19, 23, 24, 22, 26, 26, 24, 29, 29, 27, 32, 31, 30, 35, 33, 32, 38, 36, 35, 41, 38, 38, 44, 41, 39, 48, 43, 42, 51, 45, 45, 53, 48, 47, 56, 51, 50, 59, 53, 52, 62, 56, 55, 65, 58

Offset: 1

Author

Brian Barsotti, Jun 11 2019

Keywords

• nonn

Comments

Consider a square spiral which begins at the origin and spirals counterclockwise. We define (0, 0) and (1, 0) as the first two nodes. From here, we wish to find the first point along the spiral path which is closer to (0, 0) than to (1, 0). This point would be (0, 1), which is 2 unit lengths away from the previous node. Hence 2 is the first number in our sequence. Likewise, 4 is the next number, as we have to travel 4 unit lengths along the square spiral to reach a point closer to (1, 0) than to (0, 1), and so on.

Based on an empirical observation of the first several terms, I conjecture that lim_{n->infinity} a(n)/n = 8/9.

Links

• Brian Barsotti, Square Spiral Illustration
• Rémy Sigrist, Scatterplot of the first 1000 nodes
• Rémy Sigrist, PARI program for A308629

Programs

PARI

See Links section.

Extensions

More terms from Rémy Sigrist, Jun 12 2019