A384450 a(1) = 0; thereafter, a(n) is the number of arithmetic progressions of length 3 or greater at indices in an arithmetic progression ending at a(n-1).

0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 5, 0, 8, 0, 9, 0, 12, 1, 0, 1, 0, 0, 5, 0, 5, 0, 5, 1, 0, 3, 0, 3, 0, 4, 0, 2, 2, 2, 2, 3, 0, 3, 0, 2, 1, 0, 7, 0, 5, 0, 5, 0, 7, 0, 10, 1, 1, 2, 1, 1, 0, 9, 0, 6, 3, 0, 6, 1, 0, 6, 3, 3, 1, 2, 2, 3, 0, 7, 0, 6, 3, 1, 0, 4, 4

Offset: 1

Author

Neal Gersh Tolunsky, May 27 2025

Keywords

• nonn

Comments

In other words, take the longest arithmetic progression at indices with common difference k ending at a(n-1) and call that length j. a(n) is the sum of each j-2 that corresponds to a distinct common difference k. This means that an arithmetic progression of length 3 is worth 1 point, length 4 is worth 2 points, and so on.

Examples

			To find a(10), we see that there are 4 arithmetic progressions ending in a(9) = 0. These occur at indices i = 5,7,9; i = 3,5,7,9; i = 1,3,5,7,9; and i = 1,5,9. So a(10) = 4.
		

Links

• Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A308638, A362881.

Extensions

a(32)-a(86) from Pontus von Brömssen, May 30 2025