A355760 a(n) is the number of grid points in a square lattice covered by the area enclosed by n loops of an Archimedean spiral with starting point (0, 0) and endpoint (n, 0).

1, 2, 8, 21, 40, 64, 97, 132, 178, 228, 282, 350, 415, 492, 574, 660, 756, 855, 962, 1076, 1195, 1322, 1451, 1590, 1736, 1885, 2044, 2204, 2378, 2552, 2734, 2922, 3116, 3317, 3525, 3741, 3960, 4187, 4416, 4655, 4900, 5154, 5410, 5674, 5946, 6223, 6502, 6791, 7087, 7391, 7698

Offset

0,2

Comments

Grid points coincident with the outer boundary of the spiral are included.

The spiral figure is closed with a line from (n, 0) to (n-1, 0).

Conjecture: Only lattice points on the positive x-axis are on the outer boundary of the spiral. It seems that the spiral passes all other grid points without hitting any.

Links

Table of n, a(n) for n=0..50.

Karl-Heinz Hofmann, Examples of a(1..8)

Wikipedia, Archimedean spiral

Example

See the PDF in links.

Crossrefs

Cf. A000328, A295344, A046109.

Sequence in context: A000159 A358681 A090612 * A212981 A051744 A062443

Adjacent sequences: A355757 A355758 A355759 * A355761 A355762 A355763

Keyword

nonn

Author

Karl-Heinz Hofmann, Jul 16 2022

Status

approved