A308629 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

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

Table of n, a(n) for n=1..67.

Brian Barsotti, Square Spiral Illustration

Rémy Sigrist, Scatterplot of the first 1000 nodes

Rémy Sigrist, PARI program for A308629

PROG

(PARI) See Links section.

CROSSREFS

Sequence in context: A071193 A071192 A295012 * A100921 A140201 A351393

Adjacent sequences: A308626 A308627 A308628 * A308630 A308631 A308632

KEYWORD

nonn

AUTHOR

Brian Barsotti, Jun 11 2019

EXTENSIONS

More terms from Rémy Sigrist, Jun 12 2019

STATUS

approved